I basically seek a way to "bridge" the internet connection coming into my linux machine through one ethernet port (on the mother board's NIC) and share it with another computer connected to a separate ethernet port (on a NIC using PCI). In doing this, I hope to enable both computers to have internet access, using just one available internet connection. Any ideas?

The common answer of "stop being cheap, just buy a router" won't work since it's for 2 school computer for which that isn't an option. Neither computer has Wi-Fi either.