Wireless Arris Modem Won't Connect Via Ethernet

I got a new modem from the cable company Arris TG862G. This is a wireless modem and I had them set everything up on their end which is now after 2 hours with them on the phone working fine. The problem I'm having is somewhere in my computer is my guess and their guys didn't know which way was up so i gave up on them fixing my issue. So here I am. I have a custom built computer that is using a Gigabyte mobo and has been working fine with the on board Ethernet card before now. I used to have the desktop plugged into an Arris modem via a Belkin router and this set up worked fine. I also took the current modem and plugged it into the Belkin N600 router and then to my computer and to my complete bewilderment it worked fine. I wanted to get rid of the router so I could see if I could get faster download speeds (That initial issue being on speed tests I get 50mb/s and on actual downloads I would only get 5mb/s).

I got it to work (I have no idea how) for about 20 minuets. When I restarted the computer it stopped working. I am now back to square one which is the desktop not even recognizing that there is a cord plugged in. I know the cord is fine and the ports are fine as well that isn't the issue. My laptop connected just fine to the Ethernet cord with no issue. I have uninstalled the Belkin software to see if that was the issue and no change. Everything wireless works just fine, it is just this Ethernet connection that is FUBARed, and again ONLY with the desktop.I am running Windows 7 ultimate. I am about to QQ and just deal with the 5mb/s downlods because that's not terrible but I got greedy.I was messing with the Ethernet wire some and it started picking up the router but never tried to connect to the internet, I'm going to try other wires but I don't think that's the issue.

OK so another update ladies and gents: my desktop has been on for about 40 minuets now and it is connected to the modem. I have done nothing to it that would cause this and it just randomly happened. Short of never turning my computer off again which would be hell on my power bill what can I do to fix this problem? On speedtest.net I'm only getting 30.48 Mb/s and as I said earlier I should be getting around 50mb/s (which is what I get on on the wireless laptop). I just restarted it and it wouldn't connect to the modem again.