Starving the cat probably won't work. All cats are born with an instinct to hunt, even if they are well-fed. Kittens hone their hunting skills by watching their mother and playing with one another, but an only-child orphan can still be an effective mouser.
The only thing I can find to do is encourage kitty-kat when he/she catches something. Let the cat play with the "toy" as long as you can stand it (be sure to notice where it ends up.) Then praise the cat and feed it something you don't normally give it, like canned tuna.
As long as the cat is encouraged, it will keep bringing you "presents." Just be aware that cats are pretty indiscriminate when it comes to prey. Your bird / lizard / frog population is likely to take a hit as well.