The peculiar elliptical galaxy IC 1459 ( M _ { V } = -21.19 , D = 16.5 h ^ { -1 } \ > { Mpc } ) has a fast counterrotating stellar core , stellar shells and ripples , a blue nuclear point source and strong radio core emission . We present results of a detailed HST study of IC 1459 , and in particular its central gas disk , aimed a constraining the central mass distribution . We obtained WFPC2 narrow-band imaging centered on the H \alpha + [ NII ] emission lines to determine the flux distribution of the gas emission at small radii , and we obtained FOS spectra at six aperture positions along the major axis to sample the gas kinematics . We construct dynamical models for the H \alpha + [ NII ] and H \beta kinematics that include a supermassive black hole , and in which the stellar mass distribution is constrained by the observed surface brightness distribution and ground-based stellar kinematics . In one set of models we assume that the gas rotates on circular orbits in an infinitesimally thin disk . Such models adequately reproduce the observed gas fluxes and kinematics . The steepness of the observed rotation velocity gradient implies that a black hole must be present . There are some differences between the fluxes and kinematics for the various line species that we observe in the wavelength range 4569 Å to 6819 Å . Species with higher critical densities generally have a flux distribution that is more concentrated towards the nucleus , and have observed velocities that are higher . This can be attributed qualitatively to the presence of the black hole . There is some evidence that the gas in the central few arcsec has a certain amount of asymmetric drift , and we therefore construct alternative models in which the gas resides in collisionless cloudlets that move isotropically . All models are consistent with a black hole mass in the range M _ { \bullet } = 1 — 4 \times 10 ^ { 8 } \ > { M _ { \odot } } , and models without a black hole are always ruled out at high confidence . The implied ratio of black holes mass to galaxy mass is in the range 0.4 – 1.5 \times 10 ^ { -3 } , which is not inconsistent with results obtained for other galaxies . These results for the peculiar galaxy IC 1459 and its black hole add an interesting data point for studies on the nature of galactic nuclei .