  We present axisymmetric dynamical models of the edge-on S0 galaxy NGC 4342. A
combination of observations from the ground and with the Hubble Space Telescope
(HST) has shown that NGC 4342 rotates rapidly and has a strong central increase
in velocity dispersion. We construct simple two-integral Jeans models as well
as fully general, three-integral models. The latter are built using a modified
version of Schwarzschild's orbit-superposition technique. The two-integral
Jeans models suggest a black hole (BH) mass between 3 and 6x10^8 Msun,
depending on the data set used to constrain the model, but they fail to fit the
details of the observed kinematics. The three-integral models can fit all
ground-based and HST data simultaneously, but only when a central BH is
included. Models without BH are ruled out to a confidence level better than
99.73 per cent. We determine a BH mass of 3x10^8 Msun. This corresponds to 2.6
per cent of the total mass of the bulge, making NGC 4342 one of the galaxies
with the highest BH mass to bulge mass ratio currently known. The models that
best fit the data do not have a two-integral phase-space distribution function.
They have rather complex dynamical structures: the velocity anisotropies are
strong functions of radius reflecting the multi-component structure of this
galaxy. The best fit model without BH tries to fit the high central velocity
dispersion by placing stars on radial orbits. The measured rotation velocities,
however, restrict the amount of radial anisotropy such that the central
velocity dispersion measured with the HST can only be fit when a massive BH is
included in the models.
