This question is badly worded and thus difficult to answer. You should provide more details on the hardware, and how you mean "completely impersonate". Also you should confirm the nature of the router [presumably WAN] connection.
In the simple case where you are just getting an Ethernet feed and the upstream provider is not running anything fancy (like BGP for failover) the answer is "Mostly"
In this case all you need to do is clone the MAC address on the router - how you do this will depend on the OS (and if the OS does not support it, you might need to get a card which does, but this can definately be done in software under Linux). You will then need to set the same mechanisms [ie MTU, IP address, gateway, netmask] to get this to work.
Note that I said "Mostly" in my answer. This is because it is usually possible for a device to be "fingerprinted", which gives an indication of the OS. I'm not familiar with this being used very widely for network engineers (hackers are a different story). Another possibility is that the provider may have hooks [ like be running an SSH server on the router, or remote management ] which they would were no longer accessible if they provided a router to you for management purposes.
For the sake of clarity, if you are in a situation of "Provide your own CISCO router" (for example) and you want to plug in a PC directly (you will want to secure it), but make it look - to the casual observer or helpdesk dummy - who for whatever reason can directly view the remote end of the connection - they will probably just look up the first part of the MAC address and see who it was assigned to. By faking the cards mac address - which is trivial to do on ethernet - you can fool this check.