  We analyze data from the Hubble Space Telescope's Advanced Camera for Surveys
of the globular cluster Omega Cen. We construct a photometric and proper-motion
catalog using the GO-9442, GO-10252, and GO-10775 data sets. The 2.5- to 4-year
baseline between observations yields a catalog of some 10^5 proper motions,
with 53,382 high-quality measurements in a central field. We determine the
cluster center to ~1-arcsecond accuracy using two different star-count methods,
and a completely independent method using 2MASS images. We also determine the
kinematical center of the proper motions, which agrees with the star-count
center to within its uncertainty. The proper-motion dispersion of the cluster
increases gradually inwards, but there is no variation in kinematics with
position within the central ~15 arcsec: there is no dispersion cusp and no
stars with unusually high velocities. We measure for the first time in any
globular cluster the variation in proper-motion dispersion with mass along the
main sequence, and find the cluster not yet to be in equipartition. Our
proper-motion results do not confirm the arguments put forward by Noyola,
Gebhardt & Bergmann to suspect an intermediate-mass black hole (IMBH) in Omega
Cen. In Paper II we present new dynamical models for the high-quality data
presented here, with the aim of putting quantitative contraints on the mass of
any possible IMBH.
