More than 1 computer accessing internet.

I bought a Seimens 4100 DSL modem to replace a Westell 6100. I created a broadband connection on all the computers in my little home network (I have a Linksys router). I can only have internet access on one computer at a time (get a meesage that the line is busy). My old Westell allowed continuous access for all the computers without having to double click on a shortcut on the desktop to connect to the isp. How can I get all my computers to have access to the internet simultaneously?