Bump for the morning crowd, with a little more explanation on how I did it.
I placed the disc on a flat surface and put a little Wright's Brass Polish directly on the deep scratches and rubbed it in small circular motions with a soft cloth (an old t-shirt). I got pretty brave and rubbed the disc really hard. Then I rinsed off the brass polish with water, per the polish's instructions. Then I gently blotted it dry with another soft cloth (another old t-shirt). I had to repeat this about six times to get out the worst scratches. Even then, the worst scratches were still slightly visible. Also, this process leaves fine circular patterns on the disc. It definitely looks like it has been buffed.
The disc now reads normally, with no apparent problems. Just to be safe, I think I may burn off a copy.