Religious Travelers - Catholic Monasteries in French Alps

Ever since my first, and so far only, visit to Chamonix and the French Alps I have fallen in love with the place. I was thinking perhaps of heading over there next year for a bit, if possible, because I'll be staying near Toulouse for a month anyway.

I was wondering, are there any catholic (or any christian) monasteries which can accommodate the odd traveller, either for free or for a budget price? It wouldn't actually have to be in Chamonix itself, but anywhere in the French Alps would suit me just fine.

There is a good website that offers monastery stays all over Italy http://www.monasterystays.com/

If the Italian Alps will do there are lots of choices. Many of the Franch ones do also, but usually one must contact them one by one, a pain, because there seems to be no central site for them.

Not too far away, but in Switzerland is http://www.gsbernard.net/

This is not exactly what you wanted but...