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-1 (3 km s-1), 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-1 yr-1). 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^o from the plane of the sky, and is oriented at a position angle of 100^o 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-1, which corresponds to a half-opening angle of 7^o +/- 2^o, 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.