Much experimentation later... (I was curious).
Since the eSATA drive wouldn't accle and caused the PC to have a serious headache, I went after the problem in stages.
The PC has two SATA connections on the motherboard, with the two internal SATA drives connected to them. Hence the need for a PCI card, which has two more SATA connectors on the "inside" and an eSATA on the outside.
So I unplugged the second internal SATA drive from the motherboard and connected it to the PCI card, then booted up the PC. It found the drive on the PCI card with no problems at all, just as if it were connected to the motherboard.
With the HD on the PCI card, it also appeared in the boot sequence - after the graphics card and the motherboard, but before Windows started. With no drive connected to the PCI card, it didn't appear.
I removed all the drivers for the PCI card before doing the above, and let Windows 7 find the right ones (in its opinion) on the Internet.
Whereas the CD that came with the card contained "viamraid.sys", Win 7 chose "vsmraid.sys".
That indicates, I think, that the PCI card is OK and connects SATA drives with no problems.
But with an eSATA drive connected, the PC hangs, crashes, and generally gets very upset indeed under Windows 7. All that I've found on the web indicates that eSATA and Win 7 will not coexist. Lots of folks have the problem; very few have sorted it (I think they actually have their drives connected as "external SATA". I'd love to find a fix.