I went away for a day and I shut down the power to my computer and router by turning off the UPS. When I powered them back on just over 24 hours later, Vista told me "Network cable unplugged" and corresponding lights on the Gigabyte GA-P35-DS3R motherboard and the router were off. I tried restarting and shutting down the computer, I tried reinstalling the RTL8168 drivers, and I tried several cables. Nothing helped. I was beginning to think there was a hardware failure. The only hopeful sign was that the Realtek diagnostic said all tests passed and it was able to measure cable length.
Then I realized that off was not truly off for the network card, because the lights can be lit even when the computer is off. So, I powered down the computer via the switch on the back of the power supply. Then I waited a bit to be sure that the standby power capacitors discharge. When I powered up again, everything worked.
I wonder if this has anything to do with the latest F13 (2009/06/19) BIOS.