This sounds very familiar since I am having the same problem with my computer (a desktop with Intel Core i5 on a MSI motherboard). When I was still running Vista Home Premium 32bits on a Athlon 64 pc with ASRock motherboard on this connection, I had no problem connecting to the internet (through a Netgear router). But no that I am using Windows 7 64bits, my pc connects to the router but not the internet. Since I thought it might be a problem with the on board LAN, I just popped in a PCI LAN card, only to see that it has the same problem.
The only way to connect to the internet both with the on board LAN and the extra LAN card, is to switch off the LAN connection in Windows control panel and then switch it back on. Does that work for you? Of course it's not really a good solution, but at least you can go on the internet with it.