Gable's character in Dancing Lady/ Did he sing + dance in that movie?

I once saw and heard Clark Gable dance and sing on «Putting on the Ritz» in a black and white movie. I am sure it was among his first movies... I also remember that he was wearing top hat and tails and carried a stick. But which movie is the question? Is it possible that that scene was in «Dancing Lady»?

Thanks for letting me know if somebody has the answer. Bye


Francine