Are you connecting wirelessly, or using an wired ethernet connection?. I'll assume it's a wired connection, but since you didn't specifically mention such, it's good to know for sure. In addition, just to be sure it's a problem with the computer, plug in a known good computer to the modem/router and make sure the test computer works correctly.. As long as you've narrowed down that the router/modem isn't the problem,, then....
First, although you've mentioned that you installed the drivers for your hardware, make sure to visit the Dell site and download, then install any updated drivers for your ethernet card, or wireless adapter.
If it still doesn't fix the problem, here, we swap in a known good ethernet card, install the drivers, and that usually takes care of the issue. Ethernet cards go bad quite often.. I believe your computer has an ethernet port which is integrated into the motherboard but that's not a problem as there should also be some available PCI slots which you could use to install an inexpensive PCI ethernet card. It's an easy installation and should get you up and running.
Hope this helps.
I have a dell dimension 2400 that i am fixing up for a backup.
I installed a new hard drive and installed windows xp and drivers.
When i first turn on pc internet works for maybe a minute then quits. If i fix the local area connection it works for a minute then quits again.
I am hooked up to dsl.