I've tried searching for the internet for ways to do this but it only pulls up information on the actual Ethernet Shield. It seems like it should be relatively simple but I'm unsure how much software needs to be programmed on the Arduino Ethernet board, because using the ethernet from that board works just like an actual shield. Anyone have a link or advice about how to do this?
I realized my original post was unclear. I'd like to connect the Mega to the Ethernet board using the SPI pins on both and use the Ethernet hardware to connect a sketch program running on the Mega to the internet.
I don't think that you are going to be able to just borrow the hardware that way. You could connect the EthernetArduino's serial port to one of the Mega's serial ports, and have the two talk to each other that way. The EthernetArduino still needs to handle all the heavy lifting of being a server or a client, sending client requests to the Mega for handling (and for returning the server response to the client), or making client requests on behalf of the Mega, and sending server responses to the Mega.