Here's a link to a YouTube video of Carly Simon and her sister Lucy singing the song on the Phil Donahue show. Lucy wrote the tune, which was a hit in 1964, at the age of 14.

Joe, if that is the tune you are thinking of, then listening to the video will give you the definitive ending.

I think that a person who was actually singing a child to sleep with this song would need a gentler, more soothing tone than they use on the show.

https://www.youtube.com/watch?v=tzz1AZdj1LM