We present spectroscopic observations from the Hubble Space Telescope that reveal for the first time the presence of a broad pedestal of Balmer-line emission in the LINER galaxy NGC 4203. The emission-line profile is suggestive of a relativistic accretion disk, and is reminiscent of double-peaked transient Balmer emission observed in a handful of other LINERs. The very broad line emission thus constitutes clear qualitative evidence for a black hole, and spatially resolved narrow-line emission in NGC 4203 can be used to constrain its mass, with M_BH less than 6 x 10^6 solar masses at 99.7% confidence. This value implies a ratio of black-hole mass to bulge mass of less than approximately 7 x 10^-4 in NGC 4203, which is less by a factor of ~3 - 9 than the mean ratio obtained for other galaxies. The availability of an independent constraint on central black-hole mass makes NGC4203 an important testbed for probing the physics of weak active galactic nuclei. Assuming M_BH near the detection limit, the ratio of observed luminosity to the Eddington luminosity is approximately 10^-4. This value is consistent with advection-dominated accretion, and hence with scenarios in which an ion torus irradiates an outer accretion disk that produces the observed double-peaked line emission. Follow-up observations will make it possible to improve the black-hole mass estimate and study variability in the nuclear emission.
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