Thanks for your responses.
The scenario and the conclusion:
The problem started when I was using integrated graphics and had never installed a video card. To try to solve the problem, I inserted a PCI video card, connected the monitor to the video card, but still no signal. However, I could hear the hard drive churning as if the computer had successfully booted. Then I shutdown the computer, and leaving the video card in, I connected the monitor to the integrated graphics, and booted again. This time the signal reached the monitor and everything looked good. I then removed the PCI video card, and everything is still fine. My guess is that some setting on the motherboard or operating system got hosed, and the presence of a PCI video card was enough to get the setting to correct itself and pull out of the sleep mode. But that still doesn't explain why, when the BIOS was set to AUTO SELECT the video port, the motherboard still sent the video signal to the integrated output when a PCI video card was present. I'm not in the business, of correcting the idiosyncrasies of cheap motherboards, so I'll quit while I'm not too far behind.