A while back, I discovered that the rear axle on the Peugeot was bent. I ordered a new axle from Harris Cyclery, and have just now gotten everything put back together. It wasn't that hard, but life has a way of intruding on our best-laid plans. There aren't many pictures because I prefer not to leave greasy fingerprints all over the camera. Until I can hire a full-time camera operator, you'll just have to deal with it. Here's the rundown:
DISCLAIMER: I am not a professional mechanic, and this post should be read only for entertainment purposes (if you find this kind of thing entertaining). If you want to do your own work on your bicycle, you should most definitely consult other sources, and if you are not comfortable performing your own work, consult a qualified professional mechanic.
Remove the rear wheel from the frame: Without a bike stand, I just up-ended the bike and went to work. First, and most obviously, remove the quick-release skewer and make sure the little springs don't fly away. When removing the rear wheel, keep in mind that you'll have to handle the chain, as you will need to disengage it from the rear sprockets. It will be messy. I find it easiest to just remove the chain to begin with, using the chain-breaker tool, which isn't as destructive as it sounds. That way, I can give the chain a thorough cleaning, too. You don't have to do this every time you take off the rear wheel, and you probably shouldn't, but it's been a while since I did.
Once I removed the chain, the derailer "relaxed," since it's spring-loaded, and came to rest on the gear cluster. To keep the derailer out of the way, you can take a piece of wire coat hanger and rig a simple device to hold the derailer arm away from the rear sprockets. Hook one end around the derailer arm, and the other end to the frame. Wrap the wire in tape to prevent scratching the frame. With the derailer out of the way, just lift out the wheel.
Remove the old axle: For the mechanically disinclined (like me), this can be intimidating. There seem to be a lot of little parts in there, but don't sweat it too much, it's not really that bad. You should have a locknut and a bearing cone on each side of the hub (if you don't, you shouldn't be riding the bike). The bearing cone keeps the bearings contained within the hub and allows the hub to turn on the axle; the locknut keeps the bearing cone in place.
You can remove the locknut with a regular wrench, although if it's been on there a long time, you might want to put a drop or two of oil on it to make it easier to turn. There is a special wrench that makes it easy to remove and re-install the bearing cone (called a cone wrench), but I've never used one. Most sites and mechanics recommend you get one, and they're fairly cheap, so I have no excuse. Since the bearing cone shouldn't be screwed down too tight (to allow the bearings to turn freely), you can probably wiggle it loose with your fingers and a small wrench.
With the bearing cone off, the bearings will be exposed, and if they aren't contained within a ring, they will be loose within the bearing cup (the picture at left shows the bearing cup after the bearings were removed). Be very careful that you don't accidentally dump the bearings out, because if they're old and the grease is dry, they'll just fall right out, and you really don't want to lose your bearings (ba-dum-bum). On the Peugeot, each side of the hub has nine bearings (I don't know if it's different with other bikes). Once you've removed the locknut and bearing cone on one side (the side without the freewheel), you can slide the axle out of the hub. I was sitting, holding the wheel horizontally across my knees to do this, and fortunately caught the down-side bearings in my hand as I slid the axle out of the hub. The photo at left shows the hub with axle and bearings removed.
You will now be holding the old axle in one hand and the wheel in the other. Put the wheel aside for a minute. The old axle will still have the bearing cone and locknut on one side, and to get these off, you may have to grip the old axle with a pair of pliers or a vice in order to get enough leverage to turn the nut and cone off. If your axle is still good and you want to keep it, you should grip it at the center and cover whatever gripping tool you use with tape or cloth to prevent it from damaging the threads of the axle. With the final hardware removed from the old axle, set it aside or put it in the odd bits section of your toolbox.
Clean and re-grease the bearings: These literally keep your wheel turning, so if the bearings are dried up or mucky, you should clean them, as well as the bearing cup and the cone. I use Pedro's BioClean, which works quite well. Once everything is sparkly-clean, get your tube of grease and squeeze a healthy quantity around the inside of the bearing cup and spread it around with your little finger. Place each bearing back in the cup (the grease will hold them in there, but be careful handling the little buggers with greasy fingers), and when they're all in, cover them all with another dose of grease and pack in down a bit with the tip of your finger. Don't worry about putting too much in there, everything should have a nice thick coat. Do the same with the other side. You can put a piece of tape over the side you've already done just to make sure the bearings don't fall out when you turn the wheel over.
Install the new axle: I basically just did everything in reverse, threaded the bearing cone and locknut on one side of the axle, slid the axle through the hub, and threaded the cone and nut on to the other side, making adjustments to each side until the axle was centered in the hub. Again, don't over-tighten the bearing cone, the bearings need to roll within the cup. Since the freewheel is on one side of the hub, while the other is bare, you'll have to guess a little to make sure the axle is properly centered, and to do this, set the wheel back in the frame dropouts. The axle should protrude just a couple of millimeters from the frame on each side and the wheel should be centered in the frame. The new axle is a bit longer than the old one, so I added a couple of spacer washers between the bearing cone and the locknut on each side so that the quick-release could be tightened enough to firmly hold the wheel in the frame.
So, that's what I did and it seems to work just fine. For reference, the replacement axle was ordered from Harris Cyclery and has the following specs: Diameter: 10 mm; Thread: 1 mm; Length: 137 mm; Spacing: 126 mm.