Christi did a nice project here that seems just like what you are talking about, and she used hot glue:
I wouldn't recommend using hot glue, but make sure what ever you use is not used too close to the inside edge of the frame because you'll see it in the reflection of the mirror, even if it dries clear, just like you see the unpainted back side of the frame in the link posted by Steve.
I think Gorilla Glue would be your best bet.
The weight will always be a consideration. For the mirror in this project we actually screwed it into the wall being careful to place them on the OUTSIDE of the mirror: