This is Allegri’s Misereri, with the correct libretto, transcribed by FromRome.info from the recording above.

Psalm 50 in the Latin Vulgate

Miserere mei, Deus,
secundum magnam misericordiam tuam:
Et secundum multitudinem miserationum
tuarum,
dele iniquitatem meam.

Amplius lava me ab iniquitate mea:
et a peccato meo munda me.

Quoniam iniquitatem meam ego cognosco:
et peccatum meum contra me est semper.
Tibi soli peccavi, et malum coram te feci:
ut iustificeris in sermonibus tuis, et vincas
cum iudicaris.

Sacrificium Deo spiritus contribulatus;
cor contritum et humiliatum, Deus, non despicies.

Benigne fac, Domine, in bona voluntate tua Sion.
ut aedificentur muri Jerusalem.

Tunc acceptabis sacrificium justitiae, oblationes et holocausta;
tunc imponent super altare tuum vitulos.

ENGLISH from the Douay Rheims Bible

Have mercy on me, O God,
according to thy great mercy.
And according to the multitude of thy
tender mercies
blot out my iniquity.
Wash me yet more from my iniquity,
and cleanse me from my sin.
For I know my iniquity,
and my sin is always before me.
To thee only have I sinned,
and have done evil before thee:
that thou mayst be justified in thy words and mayst overcome
when thou art judged.

 A sacrifice to God is an afflicted spirit:
a contrite and humbled heart, O God, thou wilt not despise.

Deal favourably, O Lord, in thy good will with Sion;
that the walls of Jerusalem may be built up.

Then shalt thou accept the sacrifice of justice,
oblations and whole burnt offerings:
then shall they lay calves upon thy altar.