My little group and I play DnD every Friday, never the original of course, I can't stand fantasy settings. Now that the new movie came out, my Star Wars fan friend is interested in Star Trek, so I'm really wanting to get my hands on Prime Directive.
I have most of the trek RPGs people have mentioned here. The only one I ran and played was FASA Trek, which I really liked. You have to make a great deal up for the Next Gen time period but the rules are good and its depiction of Classic Trek is spot on. The others I have a few books of but never really cared for in terms of game mechanics. You could probably make Prime Directive work. The bonus here is that its really easy to utilize other GURPS stuff then. Just make sure you stay away from first edition nonGURPS Prime Directive-that's a horrible rules set designed by someo0ne who knew nothing about what makes a RPG work.