I decided to call Linksys phone help again and...believe it or not....IT'S FIXED!!
The rep had me type in 192.168.1.1 -- to go to my router configuration page.
It was set to WEP on the Security Mode.
I mentioned to the rep that I had yesterday downloaded the WPA2 patch from Microsoft for Windows XP (thanks to the FReeper suggestion) -- so I asked the rep shouldn't that setting be changed from WEP to WPA -- and he said definitely yes it should!
So he had me change it to WPA Pre-Shared Key.
Then after saving the settings, he had me click on Network Connections and pick my Wireless Connection and click "Connect".......and what do you know --- it connected!!
And it has stayed connected for several minutes now! I think I'm set!
This is great. Now I can go to the comfortable chair in the den and enjoy my new laptop.
Hopefully, this entire post will serve as a help to anyone who might have the same problem with a router.
Thanks to all for your great suggestions and interest in my efforts!
Excellent news!