Low-ionization nuclear emission-line regions (LINERs), which exist in a large fraction of galaxies, may be the least luminous manifestation of quasar activity. The nature of LINERs has, however, remained controversial because an AGN-like nonstellar continuum source has not been directly observed in them. We report the detection of bright, unresolved (FWHM $\ltorder 0.1''$) point sources of UV ($\sim 2300$ \AA) emission in the nuclei of nine nearby galaxies from a complete sample of 110 nearby galaxies imaged with {\it HST}. Ground-based optical spectroscopy reveals that five of the nuclei are LINERs, three are starburst nuclei, and one is a Seyfert nucleus. The observed UV flux in each of the five LINERs implies an ionizing flux that is sufficient to account for the observed emission lines through photoionization. The detection of a strong UV continuum in the LINERs argues against shock excitation as the source of the observed emission lines, and supports the idea that photoionization excites the lines in at least some objects of this class. Among the Northern-hemisphere galaxies in the sample, 26 are LINERs, of which only the above five LINERs have a detected nuclear UV source. There are no obvious differences in the optical line intensity ratios between the UV-bright and UV-dark LINERs. If all LINERs are photoionized, then the continuum source is unobscured along our line of sight in $5/26\approx 20\%$ of LINERs. Alternatively, spectrally-similar LINERs may be produced by various excitation mechanisms, with photoionization responsible in only about 20\% of the cases. The high angular resolution allows us to set upper limits, typically several parsecs, on the physical size of the compact star-cluster or AGN-type continuum source that is emitting the UV light in these objects.
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