Later I found mention of a 1939 recording by the Jesters. Pity that ASCAP and BMI do not date their copyrights on their search page or we could nail it down. Oh, and I found one by Rosemary Clooney from 1950. The version I can "hear" in my head featured male voices, so perhaps it was the 39.

This other stuff is all interesting too. To add to what Azizi said, our search sometimes takes us down so many different paths, all worth traveling.