We present a measurement of the systemic proper motion of the Large Magellanic Cloud (LMC) from astrometry with the High Resolution Camera (HRC) of the Advanced Camera for Surveys (ACS) on the Hubble Space Telescope (HST). We observed LMC fields centered on 21 background QSOs that were discovered from their optical variability in the MACHO database. The QSOs are distributed homogeneously behind the central few degrees of the LMC. With two epochs of HRC data and a ~2 yr baseline, we determine the proper motion of the LMC to better than 5% accuracy: μW=-2.03+/-0.08 mas yr-1, and μN=0.44+/-0.05 mas yr-1. This is the most accurate proper-motion measurement for any Milky Way satellite thus far. When combined with H I data from the Magellanic Stream, this should provide new constraints on both the mass distribution of the Galactic halo and models of the Stream.