I've connected to a wired network before and shared that over WiFi. I'm curious if I got a USB WiFi adapter if I could do the same thing with WiFi input. So the USB WiFi adapter would be the network / internet input, and the built in WiFi adapter would be the output.
If you connect to the wifi on the USB adapter and then go to the Sharing preferences pane, and enable Internet sharing using the USB adapter as the source it should work. However depending on the adapter it might not show up in Internet sharing as a source, so YMMV.
Be careful that you have permission to do this as it can be seen (at least in the U.S.) as pirating a wireless signal or some other unauthorized use.