Safari 6.2.6 no longer deals with some passwords

The college at which I teach has a staff website, <emma.morleycollege.ac.uk>, which requires a username and a password. This website and its associated webmail use a Microsoft Exchange Server — in case that is relevant to my present problem.

My operating system is Mountain Lion (10.8.5). Ever since I upgraded Safari to v. 6.2.6 I have been unable to log in. I can, sometimes by repeated entering of my details, get to a point where the home webpage is displayed, but with the dialog for username and password still in front and the webpage 'frozen', and I can get no further.

I have no difficulty logging in with Firefox. I also have Snow Leopard (Server) available in Parallels Desktop, and the Safari running in Snow Leopard allows me to log in without difficulty.

Back to Mountain Lion: I am not sure at which Safari update the change occurred, but I was certainly still able to log in with my present OS, 10.8.5 but with a somewhat earlier Safari. Can anyone suggest a solution? E.g. an earlier version of Safari that will still run in Mountain Lion?