We measured the angular rotation and proper motion of the Triangulum Galaxy (M33) with the Very Long Baseline Array by observing two H2O masers on opposite sides of the galaxy. By comparing the angular rotation rate with the inclination and rotation speed, we obtained a distance of 730 +/- 168 kiloparsecs. This distance is consistent with the most recent Cepheid distance measurement. M33 is moving with a velocity of 190 +/- 59 kilometers per second relative to the Milky Way. These measurements promise a method to determine dynamical models for the Local Group and the mass and dark-matter halos of M31, M33, and the Milky Way.