1. Buy an ethernet card for your nic-less computer. They are very cheap second hand. If you ask at a LUG close to you, someone may even give you one.
2. Use a usb-to-ethernet cable, to connect the nic-less machine to the router. You'll have to google for a howto to get this working under linux. (Note: you can basically connect any communications port to ethernet with the correct cable... serial-to-ethernet for eg. should work with linux very well.)
3. Connect the two computers via USB (or serial or parralell cables)
http://www.wown.com/j_helmig/usbmain.htm
... this concerns windows, but the principles are the same. There are other sites, google.
All these options will require you to spend money. The simplest option is #1.