Dred, if you have a Linksys router, there should actually be two DHCP events happening. Your Linksys should get its address via DHCP from your ISP, then your computer should get its address via DHCP from the Linksys. If your PC is getting a publicly routable IP address, then your router/firewall may not be doing the firewall piece very well. From the pattern of the *s you posted, your PC is probably getting a 192.168.1.xxx address, which is a private address coming from the Linksys.
There are two ways you can fix it.
1. Go into the admin web page on the Linksys and set the DHCP lease time to something really high. It should be able to go to at least a week.
or
2. Go into the settings for your computer's network card and give it a static address. Make sure the address is in the same subnet as the Linksys internal interface. The computer will get the same address every time it boots and there will be no more DHCP disconnects.