Sometimes stones get dropped or seams in the stone let go and break while carving. When this happens, the first thing to do is stop carving.
Rinse off any dust on the carving and let dry. Do not rub the pieces together, as this may alter your contact points making the breakage point more noticeable.
Once dry, apply a medium viscosity superglue on one of your contact points. Place the two pieces together, squeezing them together until excess glue weeps out the sides.
Place your carving it in a spot where it will sit undisturbed until the glue sets, usually about 15-30 minutes.
Once cured, use a rasp to remove excess glue, and continue carving.
If there is a big gap missing, you can make a filler by mixing with extra dust with an epoxy glue.