Bootes III is a disrupting dwarf galaxy associated with the Styx stellar
  stream
  We present proper motion measurements of Bo\"{o}tes III, an enigmatic stellar
satellite of the Milky Way, utilizing data from the second data release of the
Gaia mission. By selecting 15 radial velocity confirmed members of Bo\"{o}tes
III, along with a likely RR Lyrae member in the vicinity, we measure an error
weighted mean proper motion of $(\mu_{\alpha} \cos{\delta}, \mu_\delta) =
(-1.14, -0.98)\pm(0.18, 0.20)$ mas/yr. We select and present further stars that
may be Bo\"{o}tes III members based on their combined proper motion and
position in the color magnitude diagram. We caution against assigning
membership to stars that are not confirmed spectroscopically, as we demonstrate
that there are contaminating stars from the disrupting globular cluster NGC
5466 in the vicinity of the main body of Bo\"{o}tes III, but we note that our
results are consistent with previous Bo\"{o}tes III proper motion estimates
that did not include spectroscopic members. Based on the measured proper motion
and other known properties of Bo\"{o}tes III, we derive its Galactocentric
velocity and compute its orbit given canonical Milky Way potentials with halo
masses of both 0.8$\times$10$^{12}$ M$_{\odot}$ and 1.6$\times$10$^{12}$
M$_{\odot}$. These orbits robustly show that Bo\"{o}tes III passed within
$\sim$12 kpc of the Galactic center on an eccentric orbit roughly 140 Myr ago.
Additionally, the proper motion of Bo\"{o}tes III is in excellent agreement
with predictions for the retrograde motion of the coincident Styx stellar
stream. Given this, along with the small pericenter and metallicity spread of
Bo\"{o}tes III itself, we suggest that it is a disrupting dwarf galaxy giving
rise to the Styx stellar stream.
