If I understand correctly, you are trying to use a "stand-alone" Cable/DSL router in conjunction with an ADSL modem that supports an ethernet connection. If you are in the UK, this is extremely difficult to get working because UK ADSL uses PPPoA protocol (PPP over Asynchronous Transfer Mode) and Cable/DSL routers generally only support PPPoE (PPP over Ethernet). It is for this reason that the advice is always to get a *COMBINED* "Router & ADSL Modem", which does support PPPoA.
If your particular ethernet *modem* can be configured to connect automatically to the ISP when powered on (i.e. without you having to do anything in the PC to establish the connection), it should then be possible to set the router to use a "direct" connection to the "already established" internet connection in the modem.