We present measurements of positions and relative proper motions in the 30 Doradus region of the Large Magellanic Cloud (LMC). We detail the construction of a single-epoch astrometric reference frame, based on specially-designed observations obtained with the two main imaging instruments ACS/WFC and WFC3/UVIS onboard the Hubble Space Telescope (HST). Internal comparisons indicate a sub milli-arc-second (mas) precision in the positions and the presence of semi-periodic systematics with a mean amplitude of ~0.8 mas. We combined these observations with numerous archival images taken with WFPC2 and spanning 17 years. The precision of the resulting proper motions for well-measured stars around the massive cluster R 136 can be as good as ~20 microarcsec/yr, although the true accuracy of proper motions is generally lower due to the residual systematic errors. The observed proper-motion dispersion for our highest-quality measurements is ~0.1 mas/yr. Our catalog of positions and proper motions contains 86,590 stars down to V~25 and over a total area of ~70 square arcmin. We examined the proper motions of 105 relatively bright stars and identified a total of 6 candidate runaway stars. We are able to tentatively confirm the runaway status of star VFTS 285, consistent with the findings from line-of-sight velocities, and to show that this star has likely been ejected from R 136. This study demonstrates that with HST it is now possible to reliably measure proper motions of individual stars in the nearest dwarf galaxies such as the LMC.
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