Strange router/Internet connection problem

I have an edimax br-6240wg wireless router. It started giving me real trouble - the computer will tell me a network cable is unplugged, tries to get a new ip address, and then almost immediately says a cable is unplugged, ad infinitum.
Meanwhile, the wireless seems to work since other devices can connect to it, unless it'd during the time that my computer is being used and giving me all the cable unplugged problems.
So to summarize: on my computer it won't work, but the wireless works for others, unless my computer is in use.
I thought it may be the ethernet card that's the problem, but when I connect to the internet via a regular dsl modem, the internet works fine. So it doesn't seem to be the computer.
I'm wondering if it's not a virus.
Can you help?

Try factory resetting the router.

I tried both those things. I know it makes no sense. I did a reset (pressing the reset button, is that it?) and I did change the cables. Tis a mystery to me...

I just tried resetting it thru the reset function in the router settings themselves but the computer doesn't stay connected long enough to do it.

Take a look at your DHCP scope, are there enough IP's to lease in the scope?
Try plugging in a laptop with a new ethernet cable see if you can duplicate the problem with the laptop.