Sewing through the ball's stitching places more stress on the stitches than they were intended to have, and heavy-enough thread may tend to cut through them in time. If you're a little squeamish about sewing through the ball's stitching -- as I am -- here's an alternative that is less invasive:
Cut a nest hole in the mount board, smaller than the ball; maybe 1-1/2" or so.
Wrap the ball in fine mesh fabric like a lollipop. Pass the twisted end of the wrap through the nest hole, spread it out evenly around the nest hole's perimeter, pulling out any visible gathers as you go, and hotmelt glue it to the back of the mount board. No glue on the ball, of course.
The ball is supported evenly and completely by the wrap of fabric. It's easier on the hands, and I think faster than sewing through the stitches.
There are three fine mesh fabrics I know of, in order of preference: Stabilitex, 100% polyester, best and most costly (museums use it); Crepeline, imported French silk, good for most preservation mounts and costs less; nylon tulle, cheap and easily available from any good fabric store (wedding veil material), but not good for maximum preservation.
Stabilitex and Crepeline are available only from Talas (NYC), as far as I know. If anyone knows of another source, please share it.
Hope that helps.