Yeah, France at that point in time has a super weak government. Capitulation happens after losing... maybe a third of your cities? Normally it should be much harder.
The good news is that the French-italian border is not terrible to hold either. I'd start producing mountaineers and shoving them in there, as that's mainly what Italy will be fielding against you down there. Tanks and mountains do not mix.
Ah, super special pro-tip (now that I'm replaying it in order to figure out what goes where), the Italian and Extended Manginot policies add 2 to the fort level of the forts in an area- and the construction time of forts increases exponentially with how high you get them - so I'd personally avoid taking those until you've already gotten those forts up as high as you dare. I went 5 on each, but also I'm getting my teeth kicked in so who knows.
Edit: Also a lot of manual shuffling of units on the line will need to be done.
Edit 2: Also it never hurts to set a fallback line, rather than a front line. It's a little more rigid.
Edit 3: I've cut the mountaineers back to a 20-wide unit (extra-tip, multiples of 20 is best for combat width) and put another 24 troopers on that border and wow is that a messed up zone. Can definitely understand the Italy troubles.
That being said the low manpower means I've had to basically make a unit of 50-50 infantry and artillery with one anti-tank slapped on.
That's not to say it isn't possible to win as France post-capitulation (after all, the french did), but for it to be a relatively interesting game you'd want to offload a lot of your civilian construction to Algeria and SE asia.