The stellar kinematics of galactic disks are key to constraining disk formation and evolution processes. In this paper, for the first time, we measure the stellar age-velocity dispersion correlation in the inner 20 kpc (3.5 disk scale lengths) of M31 and show that it is dramatically different from that in the Milky Way. We use optical Hubble Space Telescope/Advanced Camera for Surveys photometry of 5800 individual stars from the Panchromatic Hubble Andromeda Treasury (PHAT) survey and Keck/DEIMOS radial velocity measurements of the same stars from the Spectroscopic and Photometric Landscape of Andromeda's Stellar Halo (SPLASH) survey. We show that the average line-of-sight velocity dispersion is a steadily increasing function of stellar age exterior to R=10 kpc, increasing from 30 km/s for the young upper main sequence stars to 90 km/s for the old red giant branch stars. This monotonic increase implies that a continuous or recurring process contributed to the evolution of the disk. Both the slope and normalization of the dispersion vs. age relation are significantly larger than in the Milky Way, allowing for the possibility that the disk of M31 has had a more violent history than the disk of the Milky Way, more in line with cosmological predictions. We also find evidence for an inhomogeneous distribution of stars from a second kinematical component in addition to the dominant disk component. One of the largest and hottest high-dispersion patches is present in all age bins, and may be the signature of the end of the long bar. ; Comment: Accepted for publication in ApJ. Figure 12 is the money plot