Does anyone know what Vianne's Lullaby song is called?

It's the song we first hear played by Roux on his guitar down by the river.
Vianne says: "My mother used to sing that to me when I couldn't sleep"
In a later scene Vianne is humming it partly to a sleeping Anouk, partly to her sleepless self.
I believe it's a French folk song but I cannot find the name or a recording of it ANYWHERE. It is not on the Chocolat soundtrack (boy, how I wish it was!)
Can anybody help me identify this song. I have heard it in other movies, also. I think there was a thread about in earlier in these conversations, but the name of the song was not mentioned, possibly deleted with some of the deleted posts.

Thank you!

(P.S. It is Not "Minor Swing".)