We present radial velocities for 85 cool stars projected onto the central parsec of the Galaxy. The majority of these velocities have relative errors of 1 km/sec , or a factor of 30-100 smaller than those previously obtained with proper motion or other radial velocity measurements for a similar stellar sample. The error in a typical individual stellar velocity, including all sources of uncertainty, is 1.7 km/sec . Two similar data sets were obtained one month apart, and the total error in the relative velocities is 0.80 km/sec in the case where an object is common to both data sets. The data are used to characterize the velocity distribution of the old population in the Galctic Center. We find that the stars have a Gaussian velocity distribution with a mean heliocentric velocity of -10.1+/-11.0 km/sec (blueshifted) and a standard deviation of 100.9+/-7.7 km/sec ; the mean velocity of the sample is consistent with no bulk line-of-sight motion with respect to the Local Standard of Rest. At the 1 sigma level, the data are consistent with a symmetric velocity distribution about any arbitrary axis in the plane of the sky. We find evidence for a flattening in the distribution of late-type stars within a radius of 0.4 \pc, and infer a volume density distribution of r-1/4 in this region. Finally, we establish a first epoch of radial velocity measurements which can be compared to subsequent epochs to measure small accelerations (1 km/sec yr-1), corresponding to the magnitude expected over a timespan of several years for stars nearest to Sgr A*.