European citizens don't need permits, but they can get one free if they want.
Non-European citizens need permits and they cost 119 Euros.
The French government has helpfully set up a website in French and in English where we can apply online for the new style cards. People who don't currently have permits can also use the site.
So there we are. It's all done. We'll be told when we have to pay our money and go to collect our cards.