I'm a mouse killer

Recently our cat (who has never been outside) has learned how to use the dog door. We cant exactly keep him in. It isnt worth us fighting with him. And lately, he seems to think he needs to thank us everytime he comes home. He has been bringing presents back to us in the form of mice or pack rats.
That was fine and dandy until he brought one into the house..... alive! There I was minding my own business sitting on the couch when the cat came roaming back into the house. I heard alot of commotion and looked over to see what the problem was. There was the cat, Samson, playing with the mouse! I ran over and tried to scare him into going outside.... yeah that didnt work. His thoughts: Why play outside in the cold and wind, when I can just keep on playing inside! So he ran right upstairs with this thing... still alive. I have to admit, it did have one foot in the grave... but it was still alive. I finally chased the cat away and got a dust pan and a plastic bag. I hovered over and pondered... how the hell am I supposed to kill this thing because I am not picking up a mouse that is alive to put in a bag.
I took a deep breath
I held my breath
I closed my eyes
And I jabbed the dust pan on the mouse.
Gross, I know... but it did the trick and killed the bastard. So I spent the rest of the afternoon cussing to my poor husband because his cat is doing this crap.
Needless to say, I called the vet to see if we need to worry about crawling around and Hannah. The girl at the vets office said that these mice can and do carry diseases. That if he is bringing them in the house not to let the baby around the mice if they are dead (duh!) and to put mouse traps up for the ones he brings in that are alive.
No offense but I dont like the sounds of either of these!! My solution is to get rid of the cat until the girls are old enough to know that a dead mouse is something that needs to be brought to Mommys attention right away. Right now, it really creeps me out knowing that Payton and the dog will play with it before I ever knew it was in the house. I say this cause Payton doesnt know any better and Jackson (the dog) well he is just dumb and he is a few sandwiches short of a full picnic!
So with that....... anyone want a cat?!