The Lady Is Willing(1942)

Bold, eccentric Broadway performer Elizabeth Madden befuddles her handlers by coming home with a baby she picked up on the street. She wants to keep the baby but has to find a husband to make adoption viable. She offers her new obstetrician Dr. McBain help with his research on rabbits in exchange for marriage - and he accepts. The marriage of convenience turns into a marriage of real love until Dr. McBain's ex-wife comes looking for money.