I'm a bit of a n00b about networking, so I apologize in advance. That said, if I were in your shoes, here are the things I would try:
Stop using DHCP and choose a fixed IP. Depending on how your firewall / router is set up, this may not be an issue, but I always like to test on a fixed IP until I can confirm what the issue is. If your connection works for a while and then stops working, it may be that another user is stealing your IP via DHCP. Are you on a larger network where this may occur, is it a home network? If you are the only user, this shouldn't be an issue. But I still don't like DHCP most of the time. That said, if there is poor administration on your network, you can even have IP stealing with fixed IP's. It all depends on the situation.
If the connection works for a while but then stops working periodically, see if flushing the DNS helps stability any. If it does, you could maybe make up a batch program to flush the DNS periodically.
When testing for internet connectivity, don't use the browser. It is much faster to ping in DOS. You will have to ping a known site that you know will respond, as many sites won't return packets. I always just ping Yahoo to see if I have connectivity. Also it eliminates the possibility that the problem is with the browser.
Also when using either the browser or ping to test, use the actual IP address and not the domain name. That way you can eliminate DNS as a potential issue. If you can connect with an IP but not the same domain URL for that IP, you can mostly assume that the DNS isn't resolving correctly, and therefore it may be a DNS issue.
Again, I'm not super well versed on all this, but I thought I would offer at least my two cents. It probably won't solve your problem, but it might give you some ideas. Good luck.
I must not fear. Fear is the mind-killer. Fear is the little-death that brings total obliteration. I will face my fear. I will permit it ot pass over me and through me. And when it has gone past, I will turn the inner eye to see its path. Where the fear has gone there will be nothing. Only I will remain.