Spend $40-$60 or so and get a wireless router.
Here's a place to start your search. (You want an 802.11G product; don't settle for an 802.11B product, because they're WAY slower.) Problem solved. I got a wired one (a SpeedStream 2604) for a little over $8 after rebate once-they're almost all really easy to set up and transparent to the users once you get them configured. While wireless routers tend to be a bit more work to set up, they're not at all hard to manage.
If you can't make heads or tails of the instructions that come with the router you get, chime in here and we'll walk you through it. I guarantee that this is the SIMPLEST and quickest way to get both computers online.