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We combine Gaia DR1, PS1, SDSS and 2MASS astrometry to measure proper motionsfor 350 million sources across three-fourths of the sky down to a magnitude of$m_r\sim20$\,. Using positions of galaxies from PS1, we build a commonreference frame for the multi-epoch PS1, single-epoch SDSS and 2MASS data, andcalibrate the data in small angular patches to this frame. As the Gaia DR1excludes resolved galaxy images, we choose a different approach to calibrateits positions to this reference frame: we exploit the fact that the propermotions of stars in these patches are {\it linear}. By simultaneously fittingthe positions of stars at different epochs of -- Gaia DR1, PS1, SDSS, and 2MASS-- we construct an extensive catalog of proper motions dubbed GPS1. GPS1 has acharacteristic systematic error of less than 0.3 \masyr\, and a typicalprecision of $ 1.5-2.0$\masyr. The proper motions have been validated usinggalaxies, open clusters, distant giant stars and QSOs. In comparison with otherpublished faint proper motion catalogs, GPS1's systematic error ($<0.3$ \masyr)should be nearly an order of magnitude better than that of PPMXL and UCAC4($>2.0$ \masyr). Similarly, its precision ($\sim 1.5$ \masyr) is a four-foldimprovement relative to PPMXL and UCAC4 ($\sim 6.0$ \masyr). For QSOs, theprecision of GPS1 is found to be worse ($\sim 2.0-3.0$\masyr), possibly due totheir particular differential chromatic refraction (DCR). The GPS1 catalog willbe released on-line and available via the VizieR Service and VO Service.(===GPS1 is available with VO TAP Query now, seehttp://www2.mpia-hd.mpg.de/~tian/GPS1/ for details=== )

A Gaia-PS1-SDSS (GPS1) Proper Motion Catalog Covering 3/4 of the Sky