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We present the first proper motion measurements for the galaxy M31. Weobtained new V-band imaging data with the HST ACS/WFC and WFC3/UVIS of aspheroid field near the minor axis, an outer disk field along the major axis,and a field on the Giant Southern Stream. The data provide 5-7 year timebaselines with respect to pre-existing deep first-epoch observations. Wemeasure the positions of thousands of M31 stars and hundreds of compactbackground galaxies in each field. High accuracy and robustness is achieved bybuilding and fitting a unique template for each individual object. The averageproper motion for each field is obtained from the average motion of the M31stars between the epochs with respect to the background galaxies. For the threefields, the observed proper motions (mu_W,mu_N) are (-0.0458, -0.0376),(-0.0533, -0.0104), and (-0.0179,-0.0357) mas/yr, respectively. The ability toaverage over large numbers of objects and over the three fields yields a finalaccuracy of 0.012 mas/yr. The robustness of the proper-motion measurements anduncertainties are supported by the fact that data from different instruments,taken at different times and with different telescope orientations, as well asmeasurements of different fields, all yield statistically consistent results.Papers II and III explore the implications for our understanding of thehistory, future, and mass of the Local Group. (Abridged)

The M31 Velocity Vector. I. Hubble Space Telescope Proper Motion Measurements