Forum with Single Sign On

I have a forum. I have moved the current users and posts from a very old copy of useBB, which is now obsolete, to Simple Machines Forum, choosing SMF simply because the "data moving software" was available. It can be moved from Simple Machines Forum to phpBB3 if necessary. Although, I quite like SMF. I have a custom membership system, and when members sign into the custom membership system, they must be able to visit the forum with just one click - no second login. There may be a need to add new members to the forum - or perhaps the forum will add new members as they visit the forum for the first time. You will not need to do any work on the custom membership system - I will be able to provide you with pages to post the member details from the custom membership system to the forum. The custom membership section, and the forum, are on the same server. Any forum that can accept the data will be considered. Prefence will be given to people who can show me a site where they have already done this.