Here's one I love !!!
Keren Ann isn't born French and she often sings in English but it's with French songs that she became famous in France. She started in 1998 as a leader in a band.
Here's an english one , "Not going anywhere" from the album of the same name , in the Victoires de la Musique (music awards).
http://www.youtube.com/watch?v=1504cSBhWG0
Let me know what you think about the song !