Killing a mouse in the house with a third party

I have read the question How not to kill the mouse in my house? and this got me thinking. Would it be wrong to own a cat that would then kill the mouse? I'm not referring to buying a cat strictly to kill a pest, but rather letting a prevously owned pet cat do as it pleases, and turn a blind eye when the mouse is gone. Directly, you're not the one harming the creature, and a cat killing a mouse is completely natural. Indirectly, you've caused the death of the mouse, and have reinforced the killing behavoir in the cat. Would intervening in the cat's hunting practices be good karma? The cat has plenty to eat, and the mouse only provides accomplisment to the cat.