connect the linksys via WAN/Internet port to the Uverse router.
make sure that the Linksys uses a different IP scheme. if Uverse uses 192.168.1.xxx, you need to change the LAN ip of the Linksys to 192.168.0.1
give the networks different SSIDs and change the Admin password and the password key for the wireless on the Linksys and don't give it to anybody.
also nobody should have access to the LAN ports on the Linksys since that will allow access to the rest of your network. Don't place router in a public area