Yep, that's Jack doing "Singin' in the Rain" a long time before Gene Kelly, in the finale of HOLLYWOOD REVUE OF 1929, shot in early two-color Technicolor.
The camera pans across Crawford earlier in the sequence, though, so that's not Joan standing next to Jack, although she is in the movie. Not quite sure who that is, or if she's famous - she goes by in a nanosecond in the actual film - even Jack's just a glimpse. That'd be Marie Dressler on the
other side of Jack if you watch the clip. Buster Keaton's in there too.
http://www.youtube.com/watch?v=94QR4v3zsnc