I remember this song being played daily when I lived on the south side of Chicago in the 60's. The link offered by another writer to the Marilyn May performance is of the correct song, but the version that I remember was not performed as Jazz.

I remember the lyrics and tune vividly and I'm still able to sing the chorus 58 years later. I have unsuccessfully searched for this song for many years and not knowing the name of the female artist adds to the challenge.

The lyrics I remember are slightly different than those posted by Betsy720.

CHORUS: And the rain(s) came down, It came down FROM ABOVE
HOW the rain came down, it reached above your HEAD,
But inside her shack, Sadie would entertain -
Drank coca-cola, play her Victrola,
While she enjoyed the rain.

Sigh