We present proper motions for the Large & Small Magellanic Clouds (LMC & SMC) based on three epochs of textit{Hubble Space Telescope} data, spanning a $sim 7$ yr baseline, and centered on fields with background QSOs. The first two epochs, the subject of past analyses, were obtained with ACS/HRC, and have been reanalyzed here. The new third epoch with WFC3/UVIS increases the time baseline and provides better control of systematics. The three-epoch data yield proper motion random errors of only 1-2% per field. For the LMC this is sufficient to constrain the internal proper motion dynamics, as will be discussed in a separate paper. Here we focus on the implied center-of-mass proper motions: mu_W(LMC) = -1.910 +/- 0.020 mas/yr, mu_N(LMC) = 0.229 +/- 0.047 mas/yr, and mu_W(SMC) = -0.772 +/- 0.063 mas/yr, mu_N(SMC) = -1.117 +/- 0.061 mas/yr. We combine the results with a revised understanding of the solar motion in the Milky Way to derive Galactocentric velocities: v_{tot,LMC} = 321 +/- 24 km/s and v_{tot,SMC} = 217 +/- 26 km/s. Our proper motion uncertainties are now dominated by limitations in our understanding of the internal kinematics and geometry of the Clouds, and our velocity uncertainties are dominated by distance errors. Orbit calculations for the Clouds around the Milky Way allow a range of orbital periods, depending on the uncertain masses of the Milky Way and LMC. Periods $lesssim 4$ Gyr are ruled out, which poses a challenge for traditional Magellanic Stream models. First-infall orbits are preferred (as supported by other arguments as well) if one imposes the requirement that the LMC and SMC must have been a bound pair for at least several Gyr.
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