|ADSL works through your phone lines.|
Cable works through cable.
These are two completely different media's and therefore the cablemodem will not work with the ADSL. You have to use the modem provided by the ISP.
There's a good chance you'll get a "combo" unit from the ISP. That's a device that contains the modem, a router and a wireless access point all in one unit. If that's they case, you probably won't need your linksys router.
If all you get is a modem, then you plug the network cable from it into the WAN port of the router.
It matters not how straight the gate,
How charged with punishments the scroll,
I am the master of my fate;
I am the captain of my soul.