Well, the most recent Apostolic Visit of Benedict XVI provided an excellent illustration that this is not the case. Here is the communion proper Dilexisti sung in Domplatz by a Gregorian choir.
Incidentally, they are single from a single edition of chant that is really a blown-up version from the Graduale Triplex, with the antiphon on the left and hand-written Psalms on the right.
The chant itself is certainly, then, first millennium.
"Thou lovest justice and hatest wickedness. Therefore God, thy God, hath anointed thee."
Here is a clean edition.