Axisymmetric dynamical models are constructed for the E3 galaxy M32 to interpret high spatial resolution stellar kinematical data obtained with the Hubble Space Telescope (HST). Models are studied with two-integral, f(E, L-z), phase-space distribution functions and with fully general three-integral distribution functions. The latter are built using an extension of Schwarzschild's approach: individual orbits in the axisymmetric potential are calculated numerically, and populated using nonnegative least-squares fitting so as to reproduce all available kinematical data, including line-of-sight velocity profile shapes. The details of this method are described in companion papers by Rix et al. and Cretton et al. Models are constructed for inclinations i = 90 degrees (edge on) and i = 55 degrees. No model without a nuclear dark object can fit the combined ground-based and HST data, independent of the dynamical structure of M32. Models with a nuclear dark object of mass M. = 3.4 x 10(6) M. (with 1 sigma and 3 sigma error bars of 0.7 x 10(6) M. and 1.6 x 10(6) M., respectively) do provide an excellent fit. The inclined models provide the best fit, but the inferred M. does not depend sensitively on the assumed inclination. The models that best fit the data are not two-integral models, but like two-integral models they are azimuthally anisotropic. Two-integral models therefore provide useful low-order approximations to the dynamical structure of M32. We use them to show that an extended dark object can fit the data only if its half-mass radius is r(h) less than or similar to 0."08 (=0.26 pc), implying a central dark matter density exceeding 1 x 10(8) M. pc(-3). The inferred M. is consistent with that suggested previously by ground-based kinematical data. However, radially anisotropic axisymmetric constant mass-to-light ratio models are now ruled out for the first Lime, and the limit on the dark matter density-implied by the HST data is now stringent enough to rule out most plausible alternatives to a massive black hole. The dynamically inferred M. is identical to that suggested by existing models for HST photometry of M32 that assume adiabatic growth (over a timescale exceeding 10(6) yr) of a black hole into a preexisting core. The low activity of the nucleus of M32 implies either that only a very small fraction of the gas that is shed by evolving stars is accreted onto the black hole or, alternatively, that accretion proceeds at very low efficiency, e.g., in an advection-dominated mode.
