Three french artists that immediately come to mind are Air, Frank Riggio and Lyonel Bauchet. The other two are not groups though, they are artists, but I don't really want to exclude them just because they haven't formed a band.
I love quite several tunes of theirs. 10 000Hz Legend is an awesome album, and I have particularly strong love for the Premiers Symptomes album.
Lyonel Bauchet is one that got me really excited for quite a while. He has a massive Buchla.. ahem, spaceship and he has made some really pretty tunes on it. I like to think the following song's title is addressing the Buchla for its huge range of sounds. The chirping birds are made by it.
I've been a big fan of Mylène Farmer since an early childhood and in my teenage years I liked listening to her discovery called Alizée. Farmer and her songwriting partner Laurent Boutonnat discovered her talent in the late 90s and helped her have a career.