You can go into network and sharing center and "bridge" the adapters to provide internet to the desktop. Generally though, the performance will be noticibally less then just getting a wireless card or wireless bridge for the desktop.
In order from best to worst as far as internt connectin:
1) Ethernet from router to desktop (by far the best)
2) Powerline adapter rated for 500mbps
3) Wireless bridge adapter (or repurpose another router for this)
4) Wireless card
5) Bridge accross computers
If there is only a couple of walls between the router and the desktop then a wireless bridge and a pci wireless card wil perform about the same. If they are on opposite ends of the house then the higher output power of the wireless bridge will be better.