Danny Kaye is an extra (spoilers)
Toward the end of the movie, when the men in the bar are making bets about Buck Cantrell and Ted Dillard, Danny Kaye is the one on the far left. He's got a cigar in his mouth and a bit of a French accent to boot. The camera pans across a little guy who looks like Anthony Quinn, then Danny, then to Dr. Livingson and Pres.
Alas, he's not credited, but it's definitely Danny Kaye!