  The E3 giant elliptical galaxy IC1459 is the prototypical galaxy with a fast
counterrotating stellar core. We obtained one HST/STIS long-slit spectrum along
the major axis of this galaxy and CTIO spectra along five position angles. We
present self-consistent three-integral axisymmetric models of the stellar
kinematics, obtained with Schwarzschild's numerical orbit superposition method.
We study the dynamics of the kinematically decoupled core (KDC) in IC1459 and
we find it consists of stars that are well-separated from the rest of the
galaxy in phase space. The stars in the KDC counterrotate in a disk on orbits
that are close to circular. We estimate that the KDC mass is ~0.5% of the total
galaxy mass or ~3*10^9 Msun. We estimate the central black hole mass M_BH of
IC1459 independently from both its stellar and its gaseous kinematics. Some
complications probably explain why we find rather discrepant BH masses with the
different methods. The stellar kinematics suggest that M_BH = (2.6 +/-
1.1)*10^9 Msun (3 sigma error). The gas kinematics suggests that M_BH ~
3.5*10^8 Msun if the gas is assumed to rotate at the circular velocity in a
thin disk. If the observed velocity dispersion of the gas is assumed to be
gravitational, then M_BH could be as high as ~1.0*10^9 Msun. These different
estimates bracket the value M_BH = (1.1 +/- 0.3)*10^9 Msun predicted by the
M_BH-sigma relation. It will be an important goal for future studies to assess
the reliability of black hole mass determinations with either technique. This
is essential if one wants to interpret the correlation between the BH mass and
other global galaxy parameters (e.g. velocity dispersion) and in particular the
scatter in these correlations (believed to be only ~0.3 dex). [Abridged]
