I'd start with the obvious things. Change the cable (I take it this is a simple small home network). If that does'nt work change the NIC.
It just might be the router, but obviously that's an expensive item to change - but you might need to experiment with a loaner.
If the problem moves from OS to OS I think you can pretty much discount a software/OS problem. The Linksys is a fairly well known card, and will simply install without an issue in XP, I would have thought.
I am suspicious of your network cable. Why - well the thing about network cables is that you can have a small imperfection somewhere, and if it results in an intermittent connection you might not notice until you use a bandwidth heavy application.
If you keep going on & off line, fairly rapidly your OS might be compensating by not logging you out of the network, but simply coming back online. You can remove the network cable for up to about 10 minutes in some circumstances, and still reconnect - and your PC, if it isn't doing anything, won't say a dickie bird.
Try it!