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The E3 giant elliptical galaxy IC1459 is the prototypical galaxy with a fastcounterrotating stellar core. We obtained one HST/STIS long-slit spectrum alongthe major axis of this galaxy and CTIO spectra along five position angles. Wepresent self-consistent three-integral axisymmetric models of the stellarkinematics, obtained with Schwarzschild's numerical orbit superposition method.We study the dynamics of the kinematically decoupled core (KDC) in IC1459 andwe find it consists of stars that are well-separated from the rest of thegalaxy in phase space. The stars in the KDC counterrotate in a disk on orbitsthat are close to circular. We estimate that the KDC mass is ~0.5% of the totalgalaxy mass or ~3*10^9 Msun. We estimate the central black hole mass M_BH ofIC1459 independently from both its stellar and its gaseous kinematics. Somecomplications probably explain why we find rather discrepant BH masses with thedifferent 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 athin disk. If the observed velocity dispersion of the gas is assumed to begravitational, then M_BH could be as high as ~1.0*10^9 Msun. These differentestimates bracket the value M_BH = (1.1 +/- 0.3)*10^9 Msun predicted by theM_BH-sigma relation. It will be an important goal for future studies to assessthe reliability of black hole mass determinations with either technique. Thisis essential if one wants to interpret the correlation between the BH mass andother global galaxy parameters (e.g. velocity dispersion) and in particular thescatter in these correlations (believed to be only ~0.3 dex). [Abridged]Comment: 51 pages, LaTeX with 19 PostScript figures. Revised version, with  three new figures and data tables. To appear in The Astrophysical Journal,  578, 2002 October 2

The Counterrotating Core and the Black Hole Mass of IC 1459