Try using the suggestion on this site: http://www.labnol.org/internet/disable-facebook-chat-permanently/4572/

Thank you so much for responding. Unfortunately that solution would work for the "old" facebook chat but not the new one which is much, much more invasive and difficult to get rid of.

OK, this next one is only 3 days old, so should work: http://howto.cnet.com/8301-11310_39-20080218-285/how-to-disable-facebook-chat-alerts/

Again, thank you Xircal. The solution on the CNET link didn't work for me. However I found the following information that worked for me on the second try. Here it is for anyone else with this problem:

Facebook features a great chat bar which permits you chat directly with your Facebook friends. However, this could be annoying for some users as it pops up every time users send messages. So, If you don’t want Facebook chat feature and want to disable and remove this feature permanently from your Facebook profile, here is small useful security trick to remove Facebook chat bar from your profile. The trick is very simple and brings up the use of the particular Facebook security address which, by default, does not have any chat feature.

- Login to your Facebook account.

- Now, simply add this secure Facebook address to your browser address field: https://www.facebook.com/home.php

- The whole trick here is the https which is a secure address version of the default Facebook default web address.

- Press Enter.

- You will notice that the Facebook chat bar has been removed!

(A message will appear at the top of your Facebook page that will ask you something about navigating on a secure website. Click to accept that. Now, whenever I log onto Facebook, the chat shows I am unavailable and the chat area is minimized. And I am happy about that!)