Nope, no network card needed. Both computers can be hooked together directly through network cards if you are using a cross over cable.

Another option, if you don't mind opening the computer, is take the hard drive out of the old computer and put it into the new one and copy file over that way. Then there is no network connection to slow things down.

If it is a small amount of data, you could also burn what you want to CD (if you have a burner in the old machine).