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Just an awesome idea and I wonder if anyone has done it yet.
The second half of my post is still valid though. It doesn't matter how the chat is programmed and what techniques it uses, in order to put something like that - using AJAX - on your website. So, instead of focusing your search only on AJAX chats, better look for any free open source chat.
I would highly prefer to find something written in PHP and if a DB is required MySQL.
About the only editing I could actually do would be to modify the XHTML and CSS at this moment in time.