Can't give you the chords, but you can see the sheet music at the following URL:

DRINK TO ME ONLY WITH THINE EYES at The Lester S. Levy Collection of Sheet Music.

The vocal line just above the piano part is the melody. It's done in A, maybe this will help you get the chords. I'd try to figure it out, but if it takes more than three or four chords, I'm bound to make a mess of it. - Rita Ferrara