Props the telephone tech support at Apple who treated me like I had a brain and quickly helped me to fix my problem with my broken MacBook Pro built-in iSight. He said that the following procedure fixes about 40% of built-in iSight failure cases.

1. Shut down your computer
2. Disconnect the power adaptor
3. Remove the battery
4. Press the power button for 10 seconds. Nothing visible will happen, but this is resetting something inside the computer
5. Put the battery back in
6. Attach the power adaptor again
7. Hold down the following four keys, Option-Command-P-R, and hit the power button
8. Wait until you hear the second boot-up chime noise
9. Let go of the four keys
10. Log into your user account as normal
11. Test to see if the camera works by running PhotoBooth

Thank you Apple, you did a great job!