We present high spatial resolution near-infrared spectra and images of the nucleus of Centaurus A (NGC 5128) obtained with NAOS-CONICA at the VLT. The adaptive optics-corrected data have a spatial resolution of 0.06" (FWHM) in K and 0.11" in H band, 4 times higher than previous studies. The kinematics of the ionized gas ([Fe II]) are mapped along four slit positions. The observed gas motions suggest a kinematically hot disk that is orbiting a central object and is oriented nearly perpendicular to the nuclear jet. We model the central rotation and velocity dispersion curves of the [Fe II] gas in the combined potential of the stellar mass and the (dominant) black hole. Our physically most plausible model, a dynamically hot and geometrically thin gas disk, yields a black hole mass of Mbh=6.1+0.6-0.8×107 Msolar. As the physical state of the gas is not well understood, we also consider two limiting cases: first, a cold disk model, which completely neglects the velocity dispersion but is in line with many earlier gas disk models; it yields an Mbh estimate that is almost 2 times lower. Second, a spherical gas distribution in hydrostatic equilibrium via the Jeans equation; the best-fit black hole mass increases by a factor of 1.5. This wide mass range spanned by the limiting cases shows how important the gas physics is even for high-resolution data. Our overall best-fitting black hole mass is a factor of 2-4 lower than previous measurements. A substantially lower Mbh estimate when using higher resolution kinematics was also found for many other black hole mass measurements as HST data became available. With our revised Mbh estimate, Cen A's offset from the Mbh-σ relation is significantly reduced. Based on observations collected at the European Southern Observatory, Paranal, Chile, ESO Program 72.B.0294A.