Tina, Jay J's and PB's advise are sound, but I'll give you one more option. The studs behind the drywall are going to be on 16 inch centers, now while this isn't necessarily going to line up with where you want to mount the mirror you can still take advantage of the stud(s). Take a 1"x 2" piece of wood and RIP it down the length at the middle with a 45° angle, this will give you 2 pieces that will interlock. Locate a stud (or 2 if possible) and mount one to the wall/stud(s) with the 45° pointing upwards, then mount the other piece to the back of the mirror with the 45° pointing down. The two pieces will slip into one another making for a secure fit. If you can only locate one stud to mount the wood to I'd use anchors at each end of the wood to keep it from swiveling.
Anyway, it's another option for you. Good luck! Wick