I used part of a sachet of Sugru to fix this together, but you could easily use most strong glues, especially two-part epoxy. Just remember, once you mix the parts or open the Sugru, you have a limited time to stick things together, so it's best to do all your preparation in advance:
Now move on to step 3...
Crack the eyeball in half. If the eyeball you use is like mine, you should be able to squeeze the sides to crack the seam, then gently prise it in half with a small knife blade.
Trim the scrap dowel to a length fractionally shorter than the radius of the eyeball. You should be able to stand it on your workbench, lay the half eyeball over it, and have the eyeball touch the bench all round.