A ring on which all finitely generated projectives modules are free but not all projectives are free?

Cher Michel, these rings are uncommon.

1) Over a local ring ALL projective modules are free : this is a celebrated theorem due to Kaplansky.

2) If $R$ is commutative noetherian and $Spec(R)$ is connected, every NON-finitely generated projective module is free. This is due to Bass in his article "Big projective modules are free" which you can download for free here

http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.ijm/1255637479

And now for the good news: the rings you are after are uncommon but they exist. Bass in the article just quoted shows that the ring $R=mathcal C([0,1])$ of continuous functions on the unit interval has all its finitely generated projective modules free. Nevertheless the ideal consisting of functions vanishing in a neighbourhood of zero (depending on the function) is projective, not finitely generated and not free. Bass attributes the result to Kaplansky.