  In this paper we report a new estimate of the absolute proper motion (PM) of
the globular cluster NGC 5139 ($\omega$ Cen) as part of the HST large program
GO-14118+14662. We analyzed a field 17 arcmin South-West of the center of
$\omega$ Cen and computed PMs with an epoch span of $\sim$15.1 years. We
employed 45 background galaxies to link our relative PMs to an absolute
reference-frame system. The absolute PM of the cluster in our field is:
$(\mu_\alpha \cos\delta , \mu_\delta) = (-3.341 \pm 0.028 , -6.557 \pm 0.043)$
mas yr$^{-1}$. Upon correction for the effects of viewing perspective and the
known cluster rotation, this implies that for the cluster center of mass
$(\mu_\alpha \cos\delta , \mu_\delta) = (-3.238 \pm 0.028, -6.716 \pm 0.043)$
mas yr$^{-1}$. This measurement is direct and independent, has the highest
random and systematic accuracy to date, and will provide an external
verification for the upcoming Gaia Data Release 2. It also differs from most
reported PMs for $\omega$ Cen in the literature by more than 5$\sigma$, but
consistency checks compared to other recent catalogs yield excellent agreement.
We computed the corresponding Galactocentric velocity, calculated the implied
orbit of $\omega$ Cen in two different Galactic potentials, and compared these
orbits to the orbits implied by one of the PM measurements available in the
literature. We find a larger (by about 500 pc) perigalactic distance for
$\omega$ Cen with our new PM measurement, suggesting a larger survival
expectancy for the cluster in the Galaxy.
