Is the Linksys product a router with its own inbuilt ADSL modem ? You need a model designed to work with a separate modem -- or install it instead of the existing modem.
If the router is the appropriate model, have you restarted both devices once they are connected so that they can handshake?
Is the router wireless --is there perhaps an issue between your computer's wireless and the router's?
Have you tried connecting the computer to router with an ethernet cable ?
Some more detail of the equipment and what you have tried would clarify.