You have a freewheel hub. The freewheel just screws on. It's no problem unless you have a hub with French threads. I'd replace both wheels and the freewheel. I know you can get inexpensive wheelsets with freewheel hubs somewhere online, but I can't remember where I saw them.

You must have found a good bike shop. Most will tell you that your Peugeot is obsolete junk and try to sell you a new Bike.