As with the leg repair, the arm repair can also be done by the manufacturer or perhaps be a local doll repair service. In Fig 2 below, you can see the pieces that result from a typical failure next to the repaired ball-and-socket joint. Note that the picture shows the ball-and-socket completely assembled. Doing it separately as described will help avoid getting glue inside the socket. I discovered this after the pictures were taken.
Mark the arm on the armpit side across the parting line where the half-ball meets the upper arm. Carefully cut along the parting line with an X-acto knife to remove the ball. The plastic cup may be stuck in the arm and require extraction by tapping or reaching in with a tweezer or needle-nose plier.
Use a sparing amount of super-glue to rejoin the ball the arm making sure to align the mark made earlier. A weak solution (pinch of baking soda in 3 cc (1 teaspoon) of water) may be used to speed set-up of the super-glue and eliminate stickiness of any squeeze-out.
Draw the elastic though the cup side of the ball and out the back. Draw the elastic through a second eyelet. Test the tension and compare to an attached arm . It needs to be tight enough to hold in place yet allow enough stretch to give you room to stich the ball back into the torso. When proper length is found, crimp the eyelet.
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.