Arduino Ethernet board serial connection to PC

Thank you very much for your reply, nevertheless I was wandering if it's possible to connect the 6pins directly to a Serial Port of a computer (I have a computer with a serial port) instead of using it's USB port. I thought since I have a serial port I wouldn't need the USB converter because I could connect it directly to a serial port in my PC.
I tried connecting it, but the PC doesn't recognize anything.