We present a Hubble Space Telescope (HST) study of the nuclear region of the E4 radio galaxy NGC 7052, which has a nuclear disk of dust and gas. The Second Wide Field and Planetary Camera (WFPC2) was used to obtain B, V, and I broadband images and an H alpha + [N II] narrowband image. The images yield the stellar surface brightness profile, the optical depth of the dust, and the flux distribution of the ionized gas. The Faint Object Spectrograph was used to obtain H alpha + [N II] spectra at six different positions along the major axis, using a 0." 26 diameter circular aperture. The emission lines yield the rotation curve of the ionized gas and the radial profile of its velocity dispersion. The observed rotation velocity at r = 0." 2 from the nucleus is V = 155 +/- 17 km s(-1). The Gaussian dispersion of the emission lines increases from sigma approximate to 70 km s(-1) at r = 1" to sigma approximate to 400 km s(-1) on the nucleus. To interpret the gas kinematics, we construct axisymmetric models in which the gas and dust reside in a disk in the equatorial plane of the stellar body and are viewed at an inclination of 70 degrees. It is assumed that the gas moves on circular orbits, with an intrinsic velocity dispersion due to turbulence (or otherwise nongravitational motion). The latter is required to fit the observed increase in the line widths toward the nucleus and must reach a value in excess of 500 km s(-1) in the central 0." 1. The circular velocity is calculated from the combined gravitational potential of the stars and a possible nuclear black hole. Models without a black hole predict a rotation curve that is shallower than observed (V-pred = 92 km s(-1) at r = 0."2) and are ruled out at greater than 99% confidence. Models with a black hole of mass M-. = 3.3(-1.3)(+2.3) x 10(8) M-. provide an acceptable fit. The best-fitting model with a black hole adequately reproduces the observed emission-line shapes on the nucleus, which have a narrower peak and broader wings than a Gaussian. NGC 7052 can be added to the list of active galaxies for which HST spectra of a nuclear gas disk provide evidence for the presence of a central black hole. The black hole masses inferred for M87, M84, NGC 6251, NGC 4261, and NGC 7052 span a range of a factor of 10, with NGC 7052 falling on the low end. By contrast, the luminosities of these galaxies are identical to within similar to 25%. Any relation between black hole mass and luminosity, as suggested by independent arguments, must therefore have a scatter of at least a factor of 10.