Trouble connecting to Internet each time I reboot

I recently bought an HP Elite m9500t with Vista Home Premium, SP1 (32-bit) and hooked it up to the Linksys 4-port broadband Ethernet Cable/DSL Router (Model 25-3020) that I bought in 2002. Each time I reboot, the new computer won't connect to the Internet. If I load Internet Explorer, it says "Internet Explorer cannot display the webpage." I click on "Diagnose connection problem" and I get "This computer has limited or no connectivity." It gives me the option to “Automatically get new IP settings for the network adapter,” which succeeds in connecting me to the Internet just fine—until the next time I reboot, when I have to go through the whole thing again.

Things Internet Explorer also suggested that I tried without effect: turning the modem & router off and then on again; trying the option "Reset the network adapter - Local Area Connection." I also tried disabling the DHCP BROADCAST flag as proposed at KB 928233 (http://support.microsoft.com/kb/928233/en-us), but that had no effect either. My other computer with XP SP3 continues to have no problem connecting to the Internet.

Thanks for your help!