Question : Have Router keep Internet Connection alive

I have a Linksys Wireless-B Router and a DSL Modem.
I currently have a shortcut on my desktop named Verizon DSL and whenever someone logs into the computer (Windows XP Pro SP2) they have that same shortcut that they must double click in order to connect to the Internet.  I set it up so that it automatically connects when Internet Explorer starts and needs an internet connection, but I also have another computer at another house that has a Wireless-B router and Cable modem.  On that one the router stays on the internet all the time and I don't have to click or reconnect all the time.
Is this possible with my DSL connection.

Answer : Have Router keep Internet Connection alive

Looks like you currently have a software client establishing the connection. You should be able to set up the connection on the router so that it establishes and maintains the connection. As soon as any computer is connected to the router after that it should have Internet access. To do so log on to the router (probably http:/192.168.1.1 with username admin and no password, if still default) go to the WAN/Internet connection page and enter the appropriate information. My guess would be a PPPoE connection and you need to enter your service provider's username and password. Also enable any "keep alive" feature.