The reflection of an optical wave from a metal, arising from strong interactions between the optical electric field and the free carriers of the metal, is accompanied by a phase reversal of the reflected electric field. A far less common route to achieve high reflectivity exploits strong interactions between the material and the optical magnetic field to produce a magnetic mirror which does not reverse the phase of the reflected electric field. At optical frequencies, the magnetic properties required for strong interaction can only be achieved through the use of artificially tailored materials. Here we experimentally demonstrate, for the first time, the magnetic mirror behavior of a low-loss, all-dielectric metasurface at infrared optical frequencies through direct measurements of the phase and amplitude of the reflected optical wave. The enhanced absorption and emission of transverse electric dipoles placed very close to magnetic mirrors can lead to exciting new advances in sensors, photodetectors, and light sources.