Prince has kingdom infested with mice & brings cats to remedy.

The mice are gone but the cats are now the problem. He brings dogs. Dogs become the problem so he brings lions. Lions result in elephants. Elephants need mice to them away. By the end, the prince embraces & learns to live with the mice.

I haven't heard that story before, but it sounds terribly cute. Was it an illustrated book?

I'm sorry, I should have mentioned that this is a children's illustrated book.

Is it King, the mice and the cheese?

Amazon.com: The King, the Mice and the Cheese (Beginner Books(R)) ... The story line is that first it's mice, then cats, etc., then elephants, then mice again, then ...

Sounds like it.

That's it! Thank you so much!!!