She Done Him Wrong is yet another suggestion from that 1001 Movies You Must See book and stars Mae West as the mob boss' girlfriend and Cary Grant as the head of the Christian mission next door to the bar run by the mob boss. You can download it or view it at the Internet Archive or see it here from googlevideo:
I had an awful lot of trouble with the googlevideo skipping.
Videos With Bibi has a post. 1001 Flicks has a review, as do goatdog and Film Reference.