Ok so first collect all 37 Gold Bars in the vault (might be a good idea to save the game), then speak to Elijah and get him to come to you (you shouldnt need instruction for that part). After he says he's coming down quickly turn around and leave the vault and go straight untill you reach the hallway part (or whatever) and then head for the exit where Elijah came in at. Keep walking through the hallway untill you see some crates on your left, NOW CROUCH (Elijah is already in there with you at this point) and keep moving and then turn right.
Keep going untill you reach the other hallway part (where Elijah came from), then turn right and keep going untill you come to the the second door on the left which is the door Elijah came in from. This is the exit so open the door and leave. Once you leave, your objective marker will say you trapped Elijah in the Sierra Madre vault forever, so you can stand up now and just leave. Now you're the richest man or woman in New Vegas.
