Could you provide more information on the router. Is it a NAT router? If so, then two computers can share a connection. If not, then one of the computers will need to do that.
For that, one of the computers will need two interfaces. One for the router, and the other for the LAN. You could simply connect them via a crossover cable. If you only have one interface on each computer, and the router is a NAT router, buying a switch would be easier and maybe less expensive than buying another interface.
If the router isn't a NAT router, but a modem, purchasing a Cable/DSL router would be an option as well.