In my browser at work (MSIE 6.0, Windows 2000) the DIV actually expands so that the entire line with all the w's on it will fit inside, but I see your point.

That is extremely odd actually. I assume overflow: auto; is an insatisfactory solution to the problem?
Maybe you could try and set a width to the element containing the text, since I assume there's a paragraph or another div that contains the actual chatbox entry?