Some of our American Girl dolls are 12 years old. With play and time, the elastic that holds the legs gives way and the leg is loose or falls off. While my daughter is grown, my wife still sets up a seasonal table with the girls.
American Girl provides a repair service and will re-attach arms and legs for a fee. There are doll hospitals in most large cities and individuals in smaller towns that provide doll repair. If you wish to attempt it yourself, here is one way to do it.
Undress the doll if desired. This will avoid soil and glue problems. When I examined the doll patient, I noted that the detached leg had a ball-joint that mated with a socket sewn into the cloth torso. An elastic cord is retained in the ball and holds the two parts together. The leg appears to be one piece with a hole molded in the hip-joint. There is what could be a mold-parting-line but I theorized that it was glued or sonicly welded at assembly time. I marked a small index mark on the inside of the thigh at the line with a Sharpie and carefully cut off the top of the leg with and X-acto knife. Similarly, I marked the torso-to-socket joint and cut the threads where the socket is stitched in. Inside the leg there was a plastic cup loose and a piece of elastic with an eyelet squished onto it as a retainer. In the stuffing in the torso, was a similar cup and piece of elastic. I believe that the cups are to make the elastic long enough so that it can be tensioned but still stretch far enough that the leg can be worked on. In one of the dolls, there were washers between the cup and the openings in the leg or socket. Additional strain relief? Smoother operation? Abrasion protection?
The elastic is heavy-duty shock cord about 4mm (1/8 inch) diameter. I did not have any like it but had some flat dress-makers elastic about 10 mm (3/8 inch) wide. I doubled a piece of elastic about 70 mm long (2 1/2 inches) and slipped on an eyelet with the flange away from the bitter end, drew it close and squashed the eyelet with a pliers to make a “knot”. I pushed the elastic through one of the cups so it came out the cup side. Please note that the cup in the picture Fig 2 is oriented BACKWARD. The cup should face the other way. The through the large hole in the severed leg-top. Next through the large hole in the socket, through the other cup and out the bottom. Last another eyelet is slipped on. Using another leg, estimate the tension required and pull the elastic up until the ball-to-socket tension is similar to a good joint. Mash the second eyelet to complete the assembly.
Put a small amount of cyanoacrylate glue (super glue) on the leg top where it was cut. Carefully replace the leg top matching the index you marked before cutting. Hold until set. The glue I was using remained sticky where it had squeezed out after assembly. A small amount of ammonia or a weak baking soda solution will cause this to set instantly.
Pull the leg-joint apart and to the side to give you room to work. Place the socket in the torso with the alignment correct. Use hemostats or other small nosed clamps to hold the edge of the cloth aligned with the edge of the socket. The original socket was machine stitched but I used a glover’s needle and an overhand stitch to rejoin the socket and torso. A pair of pliers may be required to force the needle through the existing holes and draw it out the other side.
When the stiching is complete around the circumference, let the leg slip into the socket and check for fit. Re-dress the doll and the job is done.
Update on Leg Repair: In one of the joints I repaired, there were a couple of washers strung on the elastic, I think possibly to keep the cord centered or reduce abrasion. They do not seem to be essential and found only in joint out of 3. Perhaps part of the original design but later dropped.
There is also a procedure for Arm Repair.