Good connection, but that version was sung by (presumably) Val Kilmer, as most of the songs from that movie were designed to evoke the musicals and spy films of the 50s and 60s.

The original version was sung by a man named Charles Hart in 1927, and there've been lots and lots of successful covers since then, by Sinatra, Elvis, Donny Osmond and Merle Haggard, and even more ... less successful covers, like the novelty album made by Joe Ross here.