Unfortunately for you, Nick is right.
You can however connect your desktop and laptop. If both desktop has two network cards, you can connect laptop to the other port and make desktop act as router. Another option would be to use bluetooth to set up network or firewire cable. You didn't tell us what OS you are using and that may be relevant. It's easier to set up Internet connection sharing on windows than GNU/Linux in my experience. Also, it would be easier to network computers via bluetooth on GNU/Linux. Windows Vista and later don't support networking over firewire, so that could be problematic.
There are also other ways to connect your laptop computer and desktop, like for example using IrDA compliant infra-red transmitters or using serial or parallel ports, but I believe that they are more trouble than they are worth. And they would probably be too slow for modern Internet connections too.
You could also try researching use of USB A to USB A cables. They, with appropriate software, can be used to connect two computers together, but I don't know if they can be used to share Internet connection.
In the end, the cheapest options are to buy a router or to buy a second network card for desktop and use it to share Internet connection. I think that if you have to buy a second network card for desktop, purchasing router instead would be better option because in order for laptop to use desktop's Internet connection, desktop would have to be on all the time.