The safest most accurate way is to cut them by hand in a miter box. Cut a piece of wood to fit snug in the base of the box and glue two blocks cut on 45 that meet at the fence. This will center the cut on the ball and be the safest way to go.

I would probably build a cradle for it by grooving out a V channel (partial cut through at two 45 degree rip bevels.) Maybe even two directions if the ball wanted to roll. Then I would use clear packing tape to secure it, put a fence on a band saw and "let 'er rip."

But there's many ways to skin this cat.

Thanks for the ideas Jey and Bobsmy. I used the packing tape to tape together two 45 degree wedges and taped the ball in the V. Where the wedges met defined the equator/cut line. I used a 1/8" bandsaw blade for finer teeth and the cut worked fine.
Thanks again.
Steve

There are many ways to skin a cat as Bobsmyuncle said. In woodworking, I believe that the most important thing a craftsman develops is not the mechanical ability to follow a premade plan. The thinking process that allows someone to design something beautiful and then figure out how to actually make it is what separates the makers and is apparent in their work. How to cut a wood ball is a small example. At least, design and problem solving in order to excute the design, are the reasons I do woodwork.

I would have stuck it in the end of an appropriately-sized PVC pipe or paper towel tube, secured it with some tape, and used it like a push stick through the bandsaw.