The priest enters the sick room itself.

The priest lays the corporal on the prepared table, places the Blessed Sacrament on it, gives the Crucifix to the sick person to kiss, and sprinkles the room with Holy Water.

Asspérges me, Dómine, hyssópo, et mundábor; lavábis me, et super nivem dealbábor,

Cleanse me of sin with hyssop, Lord, that I may be purified; wash me, and I shall be whiter than snow,

Miserére mei, Deus: secúndum magnam misericordiam tuam. Glora Patri, et Filii, et Spiritui Sancti.

Have mercy on me, O God, according to Thy great mercy. Glory be to the Father, and to the Son, and to the Holy Ghost.

The Aspérges is repeated, and the priest continues:

V. Adjútorium nostrum in nómine Dómini
R. Qui fecit caelum et terram
V. Dómine, exáudi oratiónem meam.
R. Et clamor meus ad te véniat.
V. Dóminus vobiscum.
R. Et cum spiritu tuo.
V. Orémus.

Exáudi nos, Dómine sancte, Paer omnípotens, aeterne Deus: et mittere dignéris sanctum Angelum tuum de caelis, qui custódiat, fóveat, prótegat, visitet atque deféndat omnes habitántes in hoc habitáculo. Per Christum Dominum nostrum.

V. Our help is in the Name of the Lord.
R. Who made Heaven and Earth.
V. O Lord, hear my prayer.
R. And let my cry come to Thee.
V. The Lord be with you.
R. And with thy spirit
V. Let us pray.

Hear us, holy Lord, almighty Father, eternal God: and be pleased to send Thy holy angel from Heaven to guard, cherish, protect, visit and defend all that dwell in this house. Through Christ our Lord.

The priest goes closer to the sick person and, if necessary, hears his confession.  Afterward, the Eucharist is given as it usually is outside of Mass, but the sick person, if possible, says the "Confiteor" and the "Domine non sum dignus" with the priest.

Confíteor

Confíteor Deo omnipoténti, beátæ Maríæ semper Vírgini, beáto Michaéli Archángelo, beáto Joanni Baptístæ, sanctis Apóstolis Petro et Paulo, ómnibus Sanctis, et tibi, Pater: quia peccávi nimis cogitatióne, verbo et ópere: mea culpa [strike breast] , mea culpa [strike breast] , mea máxima culpa [strike breast]. Ideo precor beátam Maríam semper Vírginem, beátum Michaélem Archángelum, beátum Joánnem Baptístam, sanctos Apóstolos Petrum et Paulum, omnes Sanctos, et te, Pater, oráre pro me ad Dóminum Deum nostrum. [the priest then says the Misereátur]

Confíteor

I confess to Almighty God, to blessed Mary ever Virgin, to blessed Michael the Archangel, to blessed John the Baptist, to the holy Apostles Peter and Paul, to all the Saints, and to you Father, that I have sinned exceedingly, in thought, word and deed: through my fault [strike breast], through my fault [strike breast], through my most grievous fault [strike breast]. Therefore I beseech blessed Mary ever Virgin, blessed Michael the Archangel, blessed John the Baptist, the holy Apostles Peter and Paul, all the Saints, and you Father, to pray to the Lord our God for me. [the priest then says the Misereátur]

Dómine, non sum dignus

Dómine, non sum dignus, ut intres sub tectum meum: sed tantum dic verbo, et sanábitur ánima mea.

Dómine, non sum dignus

Lord, I am not worthy that Thou shouldst enter under my roof; but only say the word, and my soul shall be healed.

The priest will then offer the Eucharist as Viaticum. Viaticum means "Food for the Journey" and is the Eucharist received just before death. The sick one is exempt from the Eucharistic Fast at this time (i.e., if he has eaten soon before receiving Viaticum, it is OK). When offered as Viaticum, the Eucharist is given with the following words:

Accipe, frater (soror), Viáticum Córporis Dómini nostri Jesu Christi, qui te custódiat ab hoste maligno, et perdúcat in vitam aetérnam. Amen.

Receive, brother (sister), the Viaticum of the Body of our Lord Jesus Christ; and may He keep you from the malignant foe, and bring you to life everlasting. Amen.

Prayer follows:

Dómine sancte, Pater omnipotens, aetérne Deus, te fidéliter deprecámur, ut accipiénti fratri nostro (soróri nostrae) sacrosanctum Corpus Dómini nostri Jesu Christi Fillii tui, tam córpori quam animae prost ad remédium sempitérnum: Qui tecum vivit et regnat. Amen.

O holy Lord, Father almighty and eternal God, we prayThee in faith that the holy Body of our Lord Jesus Christ, Thy Son, may profit our brother (sister) who has received it as an everlasting remedy for body and soul: Who being God, lives and reigns. Amen.

Intróeat, Dómine Jesu Christe, donum hanc sub nostrae humilitátis ingréssu, aetérna felicitas, divina prospéritas, serena laetitia, cáritas fructuósa, sanitas sempiterna effúgiat ex hoc loco accessus daemonum: adsint Angeli pacis, domumque hanc déserat omnis maligna discórdia. Magnifica, Dómine, super nos nomen sanctum tuum; et bénedic + nostrae conversatióni sanctifica nostrae humilitátis ingréssum, qui sanctus et qui pius es, et pérmanes cum Patre et Spiritu Sancto in saécula saeculórum.

O Lord Jesus Christ, let there enter this house with the entrance of our lowliness eternal happiness, divine prosperity, serene gladness, fruitful charity, everlasting health; may there fly from this place all approach of the demons; let the angels of peace be present and all ill-feeling and discord leave this house. Make Your Name great over us, O Lord, and bless + our ministry: hallow the entrance of our lowliness, Thou Who art holy, Thou who are kind, and abide with the Father and the Holy Spirit, world without end.

Orémus et deprecémur Dóminum nostrum Jesum Christum, ut benedicéndo benedicat + hoc tabernáculum et omnes habitántes in eo, et det eis Angelum bonum custódem, et fáciat eos sibi servire ad considerándum mirabilia de lege sua : avértat ab eis omnes contrárias potestates: eripiat eos ab amni formidine et ab omni perturbatióne, ac sanos in hoc tabernáculo custodire dignétur. Qui cum Patre et Spiritu Sancto vivit et regnat Deus in saécula saeculórum.

Orémus. Exáudi nos, Dómine sancte, Pater omnipotens, aetérne Deus: et mittere dignéris sanctum Angelum tuum de caelis, qui custódiat, fóveat, prótegat, visitet atque deféndat omnes habitántes in hoc habitáculo. Per Christum Dóminum nostrum. Amen.

Let us pray. Hear us, holy Lord, almighty Father, eternal God: and be pleased to send Thy holy angel from Heaven to guard, cherish, protect, visit, and defend all them that dwell in this house. Through Christ our Lord.

In nómine Patris +, et Fílii +, et Spíritus + Sancti, exstinguátur in te omnis virtus diáboli per impositiónem mánuum nostrárum, et per invocatiónem gloriósae et sanctae Dei Genitricis Virginis Mariea, ejusqye inclytu Sponsi Joseph, et ómnium sanctórum Angelelórum, Archangelórum, Patriarchárum, Prophetárum, Apostolorum, Mártyrum, Confessórum, Virginum, atque ómnium simul Sanctórum.

In the Name of the Father, and of the Son, and of the Holy Spirit, let there be extinguished in you all power of the devil by the imposition of our hands, and by the invocation of the glorious and holy Mother of God, the Virgin Mary, and of her illustrious Spouse, St. Joseph, and of all the holy Angels, Archangels, Patriarchs, Porphets, Apostles, Martyrs, Confessors, Virgins, and of all the saints together.

The priest takes the Oleum Infirmorum (the Oil of the Sick) and annoints the sick person in 6 places:
eyelids
ears
nostrils
lips
hands
feet
As he annoints each place, he says the words below.
After annointing each place, he wipes it with a piece of cotton.

Per istam sanctam Unctiónem + et suam piisimam misericórdiam, indúlgeat tibi Dóminus quidquid per (visum, audtiotum, odorátum, gustum et locutiónem, tactum, gressum deliquisti.)

By this holy unction + and his own most gracious mercy, may the Lord pardon you whatever sin you have committed by (sight, hearing, smell, taste and speech, touch, ability to walk).

The priest wipes his hands on the bread or cotton, washes them, and continues:

Orémus. Dómine Deus, qui per Apóstolum tuum Jacóbum locutus es: Infirmátur quis ín vobis? inducat presbyteros Ecclésiae, et orent super eum, ungéntes eum óleo in nómine Dómini: et orátio fidei salvábit infÍrmum, et alleviábit eum Dóminus: et si peccátis sit, remittentur ei: cura, quaésumus, Redemptor noster, grátia Sacti Spíritus languóres istius infÍrme (infÍrmae), ejusque sana vulnera, et dimitte peccáta atque dolóres cunctos mentis et  córporis ab eo (ea) expélle plenámque intérius et extérius sanitátem misericórditer redde, ut, ope misericórdiae tuae restitútus (restitúta), ad prístina reparétur offÍcia. Qui cum Patre et Spíritu Sancto vivis et regnas, Deus, in saécula saeculórum.

Lord God, Who said by Thy apsotle James: Is any man sick among you? Let him bring in the priests of the Church, and let htem pray over him, ammointing him with oil in the Name of the Lord; and the prayer of faith shall save the sick man, and the Lord shall raise him up: and if he be in sins they shall be forgiven him; we implore Thee, our Redeemer, to cure by the grace of the Holy Spirit the ailments of this sick person and heal his (her) wounds, forgive his (her) sins, drive from him (her) all pains of mind and body and in Thy mercy restore him (her) to full health within and without, that being cured by the help of Thy mercy he (she) may return to his (her) former duties, Thou Who with the Father and the Holy Spirit liveth and reigneth God, for ever and ever.

Orémus. Réspice, quaésumus, Dómine, fámulum tuum (fámulam tuam) N.____ in infirmitáte sui córporis fatiscentem, et ánimam réfove, quam creásti: ut castigatiónibus emendátus (emendáta), se tua séntiat medicina salvátum (salvátam). Per Christum Dóminum nostrum.

Let us pray. Look down, Lord, upon Thy servant (handmaid) N____ who is failing in  the weakness of his (her) body; refresh the soul created by Thee, that brought to amendment by chastisements he (she) may feel himself (herself) saved by Thy healing. Through Christ our Lord.

Dómine sancte Pater omnipotens, aetérne Deus, qui benedictiónis tuae grátiam agris infundéndo corpóribus factúram tuam multíplici pietáte  custódis: ad invocationem tui Nóminis benignus assiste; ut fámulum (fámulam) tuum (tuam) ab aegritudine liberátum (liberátam) et sanitáte donátum (donátam), dextera tua erigas, virtúte confirmes, potestate tueáris, atque Ecclesiae tuae sanctae, cum omni desideráta prosperitáte, restituas. Per Christum Dóminum nostrum.

Let us pray. O holy Lord, Father almighty and eternal God, Who by pouring the grace of Thy blessing upon the bodies of the sick tends Thy creature with many proofs of Thy love, be pleased to draw near at the invocation of Thy Name, that Thy servant (handmaid) may be delivered from sickness and endowed with health: raise him (her) up with Thy right hand, strengthen him (her) in Thy might, defend him (her) by Thy power, and restore him (her) with all desired prosperity to Thy Holy Church. Through Christ our Lord.

The Last Blessing

Orémus. Clementissime Deus, Pater misericordiárum et Deus totius consolatiónis, qui néminem vis perire in te credéntem atque sperántem: secúndum multitúdinem miseratiónum tuárum réspice propitius fámulum (fámulam) tuum (tuam) N.____ quem (quam) tibi vera fides et spes christiána commendant. Visita eum (eam) in salutári tuo, et per Unigeniti tui passiónem et mortem ómnium ei delictórum suórum remissiónem et véniam clementer indulge; ut ejus ánima in hora exitus sui te judicem propitiátum inveniat et, in sánguine ejusdem Fílii tui ab omni mácula ablúta, transire ad vitam mereatur perpetuam. Per eúmdem Christum Dóminum nostrum.

Let us pray. Most gracious God, Father of mercies and God of all consolation, Thou wish none to perish that believes and hopes in Thee, according to Thy many mercies look down favorably upon Thy servant (handmaid) N.____ whom true faith and Christian hope commend to Thee. Visit him (her) in Thy saving mercy, and by the passion and death of Thy only-begotten Son, graciously grant to him (her) forgiveness and pardon of all his (her) sins that his (her) soul in the hour of its leaving the earth may find Thee as a Judge appeased, and being washed from all stain in the Blood of Thy same Son may deserve to pass to everlasting life. Through the same Christ our Lord.

Dóminus noster Jesus Christus, Fílius Dei vivi, qui beáto Petro Apóstolo suo dedit potestátem ligándi atque solvéndi, per suam piissimam misericórdiam recipiat confessionem tuam et restituat tibi stolam primam, quam in Baptísmate recepísti: et ego facultáte mihi ab Apostólica Sede tribúta, indulgentiam plenáriam et remissionem omnium peccatorum tibi concédo. In nomine Patris +, et Fílii, et Spiritus Sancti.

May our Lord Jesus Christ, Son of the living God, Who to His apostle Peter gave the power of binding and loosing, by His most gracious mercy receive your confession and restore to you that first robe which you received at Baptism; and I, by the faculty given me by the Apostolic See, grant you a plenary indulgence and remission of all your sins, in the Name of the Father +, and of the Son, and of the Holy Spirit.

Per sacrosáncta humánae reparatiónis mystéria remittat tibi omnipotens Deus omnes praeséntis et futúrae vitae paenas, Paradísi portas apériat, et ad gáudia sempitérna perdúcat.

By the Sacred mysteries of man's redemption may almighty God remit to you all penalities of the present life and of the life to come: may He open to you the gates of paradise and lead you to joys everlasting.

Benedicat te omnipotens Deus, Pater +, et FÍlius, et SpÍritus Sancus.

May almighty God bless you, Father + and Son, and Holy Spirit. R. Amen.

Thus ends the Rite of the Sacrament of Unction.

Another thing your priest might do -- and which you should ask for -- is offer the Apostolic Blessing, which is a complete pardon for sins and for the punishment of sins if the recipient is properly disposed.

Apostolic Blessing

The primary purpose of the Sacrament is the remission of sins and the preparedness of the soul.

In the new rite, the priest asks no pardon of God for sins and the focus is on the body.
 

The matter of the Sacrament is olive oil blessed by a Bishop using these words, "Emitte, quaesumus Domine, Spiritum sanctum tuum Paraclitum de coelis in hanc pinguedinem olivae, quam de viridi ligno producere dignatus es and refectionem mentis et corporis..." ("Send forth we pray, Your Holy Spirit, the Paraclete, from heaven into this rich substance of oil").

In the new rite, any oil of plant origin may be used, blessed by a priest using these words: "May your blessing come upon all who are anointed with this oil, that they may be freed from pain and illness and made well again in body and mind and soul." The Holy Ghost is no longer invoked!
 

The form of the Sacrament is: "Through this Holy Unction or oil, and through the great goodness of His mercy, may God pardon thee whatever sins thou hast committed by evil use of sight (sight, hearing, smell, taste and speech, touch, ability to walk)."

In the new rite, it is given as "Through this holy anointing may the Lord in His love and mercy help you with the grace of the Holy Spirit. May the Lord who frees you from sin save you and raise you up." There is no request of God to remit sins!