We present the results of an internal proper motion analysis of the inner region of M15 using Hubble Space Telescope Wide Field Planetary Camera 2 images. The motions of 704 stars with brightnesses above that of the cluster turnoff (V<18.3) are analyzed to determine the velocity dispersion profile within about 15" of the cluster center. The average signal-to-noise ratio of these motions is about 3, and stars in the innermost radial bin have an average core distance of only 0.7". Assuming a distance to M15 of 10 kpc, we estimate that the velocity dispersion within this innermost bin is 14.5+/-2.5 km s-1 and that it slowly decreases outward, reaching a value of 9.8+/-0.8 km s-1 at R=15.6". We find that the proper-motion dispersion profile is in good agreement with the predictions of recent N-body simulations that do not require the presence of a central intermediate-mass black hole (IMBH). The agreement between the observed profile and the N-body simulations is marginally improved if one assumes that when a neutron star is created as a result of stellar evolution, it escapes from the cluster. Based on the results of this study and a review of other investigations, it is concluded that there is little direct evidence that M15 possesses an IMBH.