XBox Live (not 360) - setting up?

As we've just got broadband I'd like to play games online. However my PS2 doesn't have a network adapter. I ahve two affordable options:
1)Buy a network adapter
2)Sell my Ps2 and buy an XBox, which has online capabilities as standard, right?

2)Would work out costing about $0 since 2nd-hand PS2 and XBOX prices are similar in the UK. But I wanted to confirm that an XBOX really does have network adapters and stuff built in. I think that I just plug a USB cable from by ADSL modem/router into the XBOX and fire it up, creating an XBOX Live account and then am ready to go with online-playable games.
Any suggestions/advice/etc are gratefully received.

Thanks.
ps: No I'm not getting a 360 or a PS3 or a Wii.