Cats love to catch mice. And as natural hunters, they're very good at it. Your cat won't hunt because she's hungry — she's far too well fed with Hill's Science Diet™ cat food for that! She actually does it as a natural instinct. This shouldn't cause you a problem, but you should discourage her from disturbing garden wildlife. A collar with a bell will act as a warning for any unsuspecting birds, giving them the chance to fly off before the cat gets too close.
Why unwanted gifts?
On the odd occasion, your cat might also leave you a little gift of a mouse, or sometimes something bigger, on your doorstep. Cat researchers aren’t entirely sure why cats leave dead prey for their owners, but they think it's for one of two reasons:
As a sign of affection for their owner. After all, they love chasing, killing and playing with dead mice, so why shouldn't you?
As a result of a parenting instinct. Cats have a strong tendency to provide food for their family.
The manner in which cats go about their hunting could also be a factor. They like to wear their prey out, stalking and chasing until whatever they are hunting is exhausted. And they generally prefer to do this when on their home turf.
The only effective way to keep your cat from hunting is to make sure she remains an indoor cat. Of course, this may not always be practical, especially if she is accustomed to being out and about. You could try stimulating her with more play at home. Activities that simulate hunting like chasing toys or playing with other cats may give your cat the 'fix' they need to keep them from hunting outdoors.
Clean up quickly.
Remember to dispose of anything that's been caught as soon as possible. If you don't dispose of her trophies, she will think that it's OK to keep gathering them. A pair of rubber gloves might come in handy for this particular job!