Monique, Rosemary Clooney does sing the French part twice and it seems like she's repeating the same line. I don't know how good her French pronunciation was (Nat Cole's was pretty bad on some words), but here are the sounds I'm hearing:
Chantez (or "chanter")
Dansez (or "danser)
"mahn" (as in "m'en," not "mon")
"teh" or "tay" -
Here's what it sounds most like she's singing:
"Chantez, chantez m'en apprendre tes danser, danser,
Danser, dancer dans le pré.
Chantez, chantez m'en apprendre tes danser, danser,
Danser, dancer allegré."
(although if "allegré" is right, Rosie's pronunciation sounds more like "alluhgray" than "allehgray".)
Google translate reads that as
"Sing, sing to teach me your dance, dancing,
Dancing, sprightly dancing.:
That fits my admittedly limited knowledge of French, as well as the feeling of the song.
It's probably a partial mondegreen on my part, but I'm pretty sure of all of it except the "allegré" and (even less) the "apprendre tes" part.
BTW, sometimes its as hard or harder to discern the sung lyrics in one's own native language!