Radial velocities have been determined for six old open clusters in the Galactic anticenter. Moderate-resolution spectra were obtained for three to six giants in each of the clusters NGC 2141, 2158, 2204, 2243, 2420, and 2506, which yield cluster mean velocities good to about 15 km/s. These measurements are supplemented with values from the literature to provide radial-velocity information for a total of 12 open clusters with ages greater than about 2 Gyr. These data indicate that this small population of old open clusters is rotating some 25 km/s slower than the local standard of rest. The line-of-sight velocity dispersion for the cluster system is, however, much smaller than that expected for such a slowly rotating system. The relationship of the kinematics of the old open-cluster system to other disk populations is discussed.