It's called a "captive portal". You actually can run a captive portal on a Linksys Router, if its an older one that you can reflash the firmware. DD-WRT has a "hotspot" configuration that dictates a captive portal.
The problem with an encrypted hotspot--do you really plan to share the WEP/WPA key/passphrase with everyone?


)And I suppose you need a special router for that? I suppose I can't do that with my low-end Linksys router?

It sounds kind of insecure, however. The only "security" is the MAC and IP address? That sounds pretty weak!