  The 3D motions of stars in small galaxies beyond our own are minute and yet
they are crucial for our understanding of the nature of gravity and dark
matter. Even for the dwarf galaxy Sculptor which is one of the best studied
systems and inferred to be strongly dark matter dominated, there are
conflicting reports on its mean motion around the Milky Way and the 3D internal
motions of its stars have never been measured. Here we report, based on data
from the Gaia space mission and the Hubble Space Telescope, a new precise
measurement of Sculptor's mean proper motion. From this we deduce that Sculptor
is currently at its closest approach to the Milky Way and moving on an
elongated high-inclination orbit that takes it much farther away than
previously thought. For the first time we are also able to measure the internal
motions of stars in Sculptor. We find $\sigma_{R}=11.5 \pm 4.3$ km/s and
$\sigma_{T}=8.5\pm3.2$ km/s along the projected radial and tangential
directions, implying that the stars in our sample move preferentially on radial
orbits as quantified by the anisotropy parameter, which we find to be
$\beta\sim 0.86^{+0.12}_{-0.83}$ at a location beyond the core radius. Taken at
face value such a high radial anisotropy requires abandoning conventional
models for the mass distribution in Sculptor. Our sample is dominated by
metal-rich stars and for these we find $\beta^{MR} \sim 0.95^{+0.04}_{-0.27}$,
a value consistent with multi-component models where Sculptor is embedded in a
cuspy dark halo as expected for cold dark matter.
