Yes, and the theories about XBox 360s overheating are just that - theory - in my experience. I'm on my third 360 as MS replaced both the one that I bought and about a year later, the one that they replaced it with. Both locked up and went RROD upon restart within five minutes of playing in a cool room.
But David, terzaghi, I think is on his 7th 360, so I don't feel too bad.
Go to
Xbox.com and Request a Repair. I think that you need an MS Live account to register/request online.
I've found the process to be pretty cumbersome and really quite difficult before, but hopefully it'll be a smooth process for you.
