We present new kinematic measurements and modeling of a sample of 116 young stars in the central parsec of the Galaxy in order to investigate the properties of the young stellar disk. The measurements were derived from a combination of speckle and laser guide star adaptive optics imaging and integral field spectroscopy from the Keck telescopes. Compared to earlier disk studies, the most important kinematic measurement improvement is in the precision of the accelerations in the plane of the sky, which have a factor of six smaller uncertainties (sigma ~ 10 mu as yr super(-2)). We have also added the first radial velocity measurements for eight young stars, increasing the sample at the largest radii (6"-2") by 25%. We derive the ensemble properties of the observed stars using Monte Carlo simulations of mock data. There is one highly significant kinematic feature (~20sigma), corresponding to the well-known clockwise disk, and no significant feature is detected at the location of the previously claimed counterclockwise disk. The true disk fraction is estimated to be ~20%, a factor of ~2.5 lower than previous claims, suggesting that we may be observing the remnant of what used to be a more densely populated stellar disk. The similarity in the kinematic properties of the B stars and the O/WR stars suggests a common star formation event. The intrinsic eccentricity distribution of the disk stars is unimodal, with an average value of left angle bracketeright angle bracket = 0.27 + or - 0.07, which we show can be achieved through dynamical relaxation in an initially circular disk with a moderately top-heavy mass function.