Take the disc to a shop for repairs. If they can't repair it, you'll have to buy a new Black Ops.
Information on the Circular Scratch:
The Xbox 360, no matter what model (slim, original, etc) WILL eat your discs should you move it while playing. It's one of the things that just happens, and the reason why it happens is because there are no disc locks to keep the game in place. If you have a PS2, GameCube, or the original Xbox, you'll see that you have to snap the disc in place. That's so the game doesn't fly out and scratch itself.
TO PREVENT YOUR DISCS FROM GETTING THE CIRCULAR SCRATCH: place your Xbox 360 in an area where it would be almost impossible to move it. Keep it away from running children, but don't have it so high as to risk it falling down. Also, don't have the box standing up, leave it on it's side. Neither side will completely stop the disc from scratching, but leaving it horizontal will make it less prone from tipping.