  We present new proper motions from the 10 m Keck telescopes for a puzzling
population of massive, young stars located within 3.5" (0.14 pc) of the
supermassive black hole at the Galactic Center. Our proper motion measurements
have uncertainties of only 0.07 mas/yr (3 km/s), which is ~7 times better than
previous proper motion measurements for these stars, and enables us to measure
accelerations as low as 0.2 mas/yr^2 (7 km/s/yr). Using these measurements,
line-of-sight velocities from the literature, and 3D velocities for additional
young stars in the central parsec, we constrain the true orbit of each
individual star and directly test the hypothesis that the massive stars reside
in two stellar disks as has been previously proposed. Analysis of the stellar
orbits reveals only one of the previously proposed disks of young stars using a
method that is capable of detecting disks containing at least 7 stars. The
detected disk contains 50% of the young stars, is inclined by ~115 deg from the
plane of the sky, and is oriented at a position angle of ~100 deg East of
North. Additionally, the on-disk and off-disk populations have similar K-band
luminosity functions and radial distributions that decrease at larger projected
radii as \propto r^-2. The disk has an out-of-the-disk velocity dispersion of
28 +/- 6 km/s, which corresponds to a half-opening angle of 7 +/- 2 deg, and
several candidate disk members have eccentricities greater than 0.2. Our
findings suggest that the young stars may have formed in situ but in a more
complex geometry than a simple, thin circular disk.
