If you ask the new insurer they should do all the necessary (cancel your existing contract etc) for you. You might need to ask your existing insurer for a releve d'information but even that, ime, has been handled by the new insurer.
If not, there are template letters that you can find on the web to send and, yes you're right, it's a month before. Your insurer you should send you a renewal notice latest a month before renewal date and you can cancel once you've received that.
Tbh, I think it's much simpler than it once was but I am going back many years now (10+)
Edit: Links to template letters....
https://www.service-public.fr/particuliers/vosdroits/F2659
https://www.service-public.fr/particuliers/vosdroits/R2689