As a step toward elucidating the physical conditions in nearby active galaxies, this paper presents spectral energy distributions (SEDs) of a sample of seven low-luminosity active galactic nuclei (AGNs). SEDs for four objects are presented for the first time (NGC 4261, NGC 4579, NGC 6251, and M84); the data for the remaining three (M81, M87, and NGC 4594) have been substantially updated compared to previous studies. The nuclear fluxes were carefully selected so as to avoid contamination by emission from the host galaxy, which can be substantial for very weak nuclei. The present sample of low-luminosity nuclei all exhibit SEDs that look markedly different from the canonical broad-band continuum spectrum of luminous AGNs. The most striking difference is that the low-luminosity objects lack an ultraviolet excess (the "big blue bump"), a feature normally associated with emission from a standard optically thick, geometrically thin accretion disk. The weakness of the ultraviolet band leads to an unusually steep optical-ultraviolet continuum shape and a more pronounced contribution from the X-rays to the ionizing spectrum. It is argued that the absence of the big blue bump is a property intrinsic to the SEDs and not an artifact of strong dust extinction. The integrated spectra confirm the exceptional weakness of the nuclei: the bolometric luminosities range from 2E+41 to 8E+42 erg/s, or 10^-6 to 10-3 times the Eddington rate for the black hole masses previously reported for these galaxies. (Abridged)

To appear in The Astrophysical Journal. LaTex, 34 pages including embedded figures and tables