3M Auto Exterior Attachment Tape, 1/4 in x 5 ft, Heavy Duty Double-Sided Automotive Tape for Car Trim, Moldings, Emblems, and Badges (38582)

I second the 3M tape like this It’s designed for external auto use and will hold the mirror in place. Scrape the old residue from the glass and the rear view mirror, then clean it, let it fully dry, then apply the tape.