The elastic that is used to attach a Madame Alexander doll's head and limbs to the torso will, like any doll elastic, break down over time and become stretched out or break. You can repair your doll by restringing it using new doll elastic and a minimal amount of tools. You will need some patience as it might take more than one attempt to get the doll properly strung and to get the tension of the elastic just right.
Examine how the the elastic has been strung to hold the doll together before removing any loose elastic. This may not be possible if the doll's limbs are no longer attached to the body.
Cut and remove the old elastic. All old and stretched out elastic should be completely removed from the doll's body.
Cut enough new elastic to properly string together the doll in its original configuration while also allowing enough elastic to form knots.
Use your fingers or a doll stringing hook to thread the elastic through the loop of one limb. Leave one end of the elastic outside the torso. Thread the other end through the torso then through the loop on the opposite limb. Thread the elastic back through the torso to connect other limbs or back through the torso to the elastic end if only two limbs are being connected.
Use a hemostat or your fingers to pull both ends of the elastic tight. Join together the two ends of the elastic in a secure knot. Trim the ends of the knot.