Use a utility knife to make a diagonal cut on the belt at the exact point it has broken. Each side needs an opposite angle cut so that when placed on on top of the other, the angle conforms to a symmetrical looking round belt. It does not have to be exact, just close so that they fit together.
Smear some goop on the cut sides.
Splice the sides together, goop side to goop side.
Thread the needle with cotton thread and tie off the ends, just like in regular sewing.
Insert the needle through the center of the belt about 1/16 of an inch from the end of the splice. Pull it all the way through until the knot catches.
Wrap the cotton thread around the rubber belt making your way over the entire splice. Wrap the belt firmly.
Push the needle through the center of the rubber belt when you reach 1/16 inch past the other side of the splice, and tie it off. If you have pulled the thread tightly, there will be a slight indentation in the rubber that the thread covers.
Smear more goop over the top of the thread covering it completely, even over the knots, and attempt to fill in the indentation as close to level as possible with the goop. Let dry for 24 hours.
Wait 24 hours, cut the knots off with your utility knife and your belt is ready.