The way it wasn't working was that it saw a strong signal, it got an IP address and other stuff from the DHCP of the router, but IP from or to it just wouldn't work. So I updated the router firmware to the latest, installed the latest OSX and Airport software, and still it wouldn't work. What was most wierd is that when I plugged in an EtherNet connection, then both the EtherNet and the Wireless connection worked, but as soon as I unplugged the EtherNet the Wireless stopped working too. OSX has pretty good networking to allow this sort of dynamic change, but I thought it might be the culprit.
Then I changed the Beacon Interval (what ever that is) on the Router from 100 to 50 msec which someone in comp.sys.mac.comm had reported that LinkSys had advised but hadn't worked for them. And lo and behold, the Wireless connection started working. I plugged and unplugged the EtherNet cable a couple of times and everything still worked correctly.
Then I went out for a while to get a haircut and when I came back there was no Wireless Signal on the Mac. It turns out that Robert had to reboot the router since ATTBI had changed something so that DNS didn't work; something that happens all too periodically. So I changed the Beacon Interval back from 50 to 100 and now everything is working fine!
Makes no sense.