Forum for Logged In Users Only


I have a members only section on my website (which is not hosted through or com) that has a separate navigation menu which only appears when users are logged in.

I would like to add a forum to this but I don't want my users to login twice - ie to the web site and then to the forum. Also I don't want the forum to be able to be accessed (or searched by google) by a non member. The forum should be able to show images that users can upload.

Any advice for the best forum software to do this will be most gratefully received. Many thanks.