So, after the very nice Time Warner service man called, Trusty Desktop ran fine til bedtime. This morning, I powered him up, and lo and behold, it was just as broke, and broke the same way, as before the service guy arrived. Same deal, the laptop (Angry Pierre) worked and Trusty Desktop didn't. After a couple of hours of messing around, I finally disabled the Zone Alarm firewall. Presto, internet came back.
So, I used Add/Remove Programs to zap Zone Alarm for good. Turned on the less effective built in Windows firewall. In actual fact the Netgear router is a very good firewall, and hackers/crackers/spyware and such get stopped by the router. I checked the Zone Alarm logfile and it hadn't seen an attack since it had been installed. So, for now, bye bye Zone Alarm. Been running Zone Alarm for years and years, without trouble, but looks like it's broke now.