The cable modems used here in the UK by Virginmedia will only give out an IP address to 1 or 2 MAC addresses. I have seen this on a family relations setup. The cable modem would give out an IP address to their Laptop, but not to their router. What I had to do was clone the MAC address on the router, so that to the cable modem thought it was giving the IP address to the same machine.
Some times, and only some times, power cycling the cable modem will make it give the address to new hardware.
If you do not want to get a router then get another ethernet connection on the PC that does connect and turn on ICS (Internet connection sharing) I have not done this so I do not know how well it works.
Unless funds are realy tight get a wired or wireless router. Most of the home routers have a 4 port switch built into them so you could sell the GS605 to offset the cost. As you only have 2 machines and 1 of them has a 10/100 BT ethernet card then the Gigabit functionality is of no use at present.
If you do get the router you will need to set its MAC address on the WAN port to be the same as the MAC address of the PC that connects at present. This is what is meant by "cloning". The router will significatly reduce the amount of malware and nasties your machines will get from the net.
I do not know what prices the routers are in the US, but I'm sure some else here can advise.
Rob Murphy.