  We report the discovery of an extremely broad H-alpha emission line in the
LINER nucleus of NGC 4579. From ground-based observations, the galaxy was
previously known to contain a Type 1 nucleus with a broad H-alpha line of FWHM
= 2300 km/s and FWZI ~ 5000 km/s. New spectra obtained with the Hubble Space
Telescope and a 0.2 arcsec-wide slit reveal an H-alpha component with FWZI ~
18,000 km/s. The line is not obviously double-peaked, but it does possess
shoulders on the red and blue sides which resemble the H-alpha profiles of
double-peaked emitters such as NGC 4203 and NGC 4450. This similarity suggests
that the very broad H-alpha profile in NGC 4579 may represent emission from an
accretion disk. Three such objects have been found recently in two HST programs
which have targeted a total of 30 galaxies, demonstrating that double-peaked or
extremely broad-line emission in LINERs must be much more common than would be
inferred from ground-based surveys. The ratio of the narrow [S II] 6716, 6731
lines shows a pronounced gradient indicating a steep rise in density toward the
nucleus. The direct detection of a density gradient within the inner arcsecond
of the narrow-line region confirms expectations from previous observations of
linewidth-critical density correlations in several LINERs.
