In the recording by the University Six (which you can hear at the Red Hot Jazz Archive—see Alan's link above) only the chorus us sung. I listened, and here's what I think I hear:

CHORUS: He plays those beautiful changes in diff'rent keys,
Beautiful changes and harmonies.
He starts in B, then changes to D.
He's fooling around, most any old key
What's that? Hear that minor strain.
That's it! There's his note again.
There's so many babies that he can squeeze,
And he's only changing those keys