Hmm. Was the version of the controller driver that you download from the motherboard maker (
http://www.gigabyte.com/products/product-page.aspx?pid=3434#dl ) and have you installed it yet? If so, and the controller is still flagged as bad, I would try, as a last-ditch effort, uninstalling the device in Device Manager and rebooting. When it finds new HW, point it to the recently downloaded driver. If that still doesn't work, get an eSata drive and forget about USB.
Your PC has three-way graphics and a floppy drive? "Something old, something new."
----------------------
Have you tried the drive on USB ports from the chipset, not the NEC 3.0 ports? Your spec says:
Integrated in the South Bridge:
1. Up to 12 USB 2.0/1.1 ports (6 on the back panel, including 2 eSATA/USB Combo, 4 via the USB brackets connected to the internal USB headers)
NEC chip:
1. Up to 2 USB 3.0/2.0 ports on the back panel
I have no idea what a USB / eSATA combo port is. None.