So it turns out that the new ethernet card did not fix my problem. My desktop would still swich networks, cutting out my internet connection, even though it is connected by a cable. I had an epiphany that it was not hardware related, but software. My laptop, which is wireless is doing fine. Liz's computer, which is connected by a cable is doing fine. Only my desktop was wonky. Also when I connected to the old router everything was fine.
So it occured to me that the only difference between my computer and Liz's computer is that I installed the software that came with our wireless router. Thus, I felt, my desktop was thinking it was a wireless machine and would occasionally go looking for a wireless network. But having no wireless card it got confused.
So I uninstalled the router software and voila! No more problems.