Your Peugeot almost certainly has French cups (if they're stamped 35 x 1 you have confirmation), and if you can save/service them, I'd recommend it. There was a recent thread on this forum discussing French BBs - you should refer to that.
The spindle tends to pit before the races, so perhaps you can find a replacement spindle and put in new bearings. If so, you should leave the fixed cup in the frame. French fixed cups thread in clockwise. That means that the bearing spin direction tends to loosen the fixed cup as you pedal. If the cup is presently in good and tight, leave it that way.
If you must remove it, you'll either need the proper wrench (see Park Tool) or, preferably, clamp the BB flats in a good bench vice and carefully turn the frame.