Using the Very Large Array with an angular resolution of approximately 3'', we have detected the hydrogen recombination line H92alpha (nu(rest) = 8309.38 MHz) from the starburst nuclei in the galaxies NGC 3628 and IC 694 and also from the Seyfert II nucleus in NGC 1365. In each case the line-emitting region extends over a few hundred parsecs. The detected lines have peak intensities in the range 0.5-1.5 mJy and widths (FWHM) 200-400 km s-1. The line was not detected in three other galaxies (NGC 262, NGC 1068, and NGC 3079) that were observed to a similar sensitivity level. We present a model in which a collection of H ii regions in the nuclear region accounts for the observed H92alpha line. The required number of H II regions, their temperature, electron density, and linear size are constrained by the observed line flux density, line width, continuum spectrum, and size of the line-emitting region. If the temperature of the H II regions is above 5000 K, then electron densities in the range 5-50 x 10(3) cm-3 are permitted by the available constraints. Several hundred H ii regions of a few parsecs in size, with a total mass of a few times 10(5) M., are required to account for the observed line flux density. The rate of production of Lyman continuum photons required to maintain the ionization is a few times 10(54) s-1. Much of the line emission comes from internal stimulated emission due to the continuum generated within the H II regions which account for 5%-30% of the observed total continuum at 5 GHz. Predictions are made for the expected recombination line and continuum flux density as a function of frequency from the nuclear region of the galaxies.