We present long-baseline Faint Object Spectrograph (FOS) spectra of the nuclear accretion disk in M87 (NGC 4486), offset from the nucleus by 0.6" (42.7 pc) in order to avoid the nuclear continuum. Even so close to the nucleus, the optical spectrum has the appearance of a normal LINER galaxy. We show that the presence of strong UV emission lines provides a definitive test of the excitation mechanism; the disk is shock excited, not photoionized by a UV continuum from the central source. The shock velocity inferred (265 km s-1) is about one-half of the Keplerian rotation velocity found earlier by Ford et al. Since shock dissipation appears to be the principal means of increasing the binding energy of the accreting gas, we can use the FOS data and the luminosity profile of the accretion disk to estimate the rate of mass accretion as a function of radius. We find that this rate decreases with decreasing distance from the nucleus, as the material becomes organized into a cool and thin classical accretion disk in the inner regions. In the outer disk, the accretion rate (~4 M⊙ yr-1) is comparable to that determined for the X-ray-emitting cooling flow, showing that a large fraction of the cooling gas can find its way into the nuclear regions. The accretion rate near the nucleus (~3 × 10-2 M⊙ yr-1) is consistent with the properties of the relativistic jet and its associated radio emission. Over the lifetime of the jets, about 107 M⊙ of cool material may have accumulated in the nuclear regions, allowing the formation of a disk that is optically thick to Thomson scattering where it becomes ionized close to the nucleus. We speculate that LINER emission is a general property of the shocked dissipative regions of accretion disks in active galaxies with strongly sub-Eddington accretion and may therefore be used as a diagnostic of these dissipative accretion flows.