Problem Solved!
I replaced my USB2Serial cable and the problem appears to be gone (36 hours so far with no issues).
For the record:
Bad: USB-2-Serial cable with the Prolific chipset and latest drivers (from both the Prolific web site and cable manufacturer web site). Did not work reliably on my Win 7-64 system.
Good: USB-2-Serial cable with the FTDI chipset, and whatever drivers Windows found. (Note that I did not install any drivers myself. I simply plugged in the cable and let Windows search for the appropriate drivers, which it was able to find.)
For those of you who are interested, you can find USB-2-Serial cables with the FTDI chipset at
www.usbgear.com. Do a search for "FTDI". Also, for many of their cables it says which chipset is used in the product descriptions.
Thanks everyone,
Matt