And the RAID array reported that one of the drives had failed. Happily, I had a spare on the shelf, as I'd bought two drives during the last disk failure -- the first on-line, the second from a local store when I decided not to wait to restore the mirror. So I popped in the replacement drive and it's happily getting rebuilt now.
Interestingly, the failed drive is still under warranty according to the Seagate site. But they suggest testing it with SeaTools first before sending it back, as they'll bounce it if it passes their tests and charge you for shipping. So the SeaTools tests are happily running on the drive in a USB carrier and finding nothing wrong.
Mind you, Windows refuses to actually recognize the drive, but why should that mean that it isn't working?
Update: I just checked. This computer is nearly six years old now. I don't know when I've had a six year old computer still running, as usually I replace them on a shorter cycle. It's just not clear that I'd actually improve performance significantly by going to a new build and reinstalling everything is a real pain.