The solution: Hook and loop fasteners! These happen to be Velcro brand, but any hook-and-loop fasteners with adhesive backing should work. I bought this package from Home Depot, but I'm sure any hardware/home improvement store will carry this stuff.
Cut to size and attach hook and loop fasteners -- stick one piece on the helmet, and attach the other piece to the visor, ensuring the two pieces line up.
All happy! Your visor will now stay in place. My job is a little sloppy; I'm sure you can do better.