OK, I have been able to fix the problem, but I am not exactly sure why! I removed both the existing, original TSST DVD drives (1 ROM and 1 RW). The DVD-ROM was on SATA1 and the DVD/RW was on SATA2. I removed them and went into the BIOS and turned off those ports. The only port in use was SATA0 for the hard drive. I re-booted and then shut down the PC. I then attached the new, Optiarc DVD/RW as SATA1. On the reboot, I went into the BIOs to turn on the port and verify that it recognized the Optiarc. It did! I then shut down again and connected the TSST DVD/RW as SATA2. On re-boot I again went into the BIOS and turned on this port and verified that the drive was recognized. After saving the settings, I went into windows and everything is working. I still don't know why, except that I did each step individually rather than in one step. Anyway, don't want to "look a gift horse in the mouth"!
Thanks to those who took the time to assist and suggest.