Three galaxies known to be LINERs were observed spectroscopically in the ultraviolet in an attempt to detect the presumed nonthermal continuum source thought to be the source of photoionization in the nuclei. NGC 4501 was found to be too faint for study with the IUE spectrographs, while NGC 5005 had an extended ultraviolet light profile. Comparison with the optical light profile of NGC 5005 indicates that the ultraviolet source is distributed spatially in the same manner as the optical starlight, probably indicating that the ultraviolet excess is due to a component of hot stars in the nucleus. These stars contribute detectable absorption features longward of 2500 A; together with optical data, the IUE spectra suggest a burst of star formation about 1 billion yr ago, with a lower rate continuing to produce a few OB stars. In NGC 4579, a point source contributing most of the ultraviolet excess is found that is much different than the optical light distribution. Furthermore, the ultraviolet to X-ray spectral index in NGC 4579 is 1.4, compatible with the UV to X-ray indices found for samples of Seyfert galaxies. This provides compelling evidence for the detection of the photoionizing continuum in NGC 4579 and draws the research fields of normal galaxies and active galactic nuclei closer together. The emission-line spectrum of NGC 4579 is compared with calculations from a photoionization code, CLOUDY, and several shock models. The photoionization code is found to give superior results, adding to the increasing weight of evidence that the LINER phenomenon is essentially a scaled-down version of the Seyfert phenomenon.