We present photometric and spectroscopic observations of SN 1991bg, a very peculiar Type Ia supernova located in the outskirts of the E1 galaxy NGC 4374 (M84) in the Virgo cluster. At maximum brightness SN 1991bg was 1.6 mag subluminous in V and 2.5 mag subluminous in B, compared with normal SNe Ia. The colors were unusually red, but the object was not significantly reddened by dust. The decline from maximum was certainly quite steep; we measure an initial linear V fading of 0.10 mag d^-1^ rather than the typical value of 0.06 mag d^-1^ for SNe Ia, and a late- time decline of 0.034 mag d^-1^ rather than 0.026 mag d^-1^. Furthermore, the knee in the Flight curve of SN 1991bg occurs only ~17 d past B maximum, in contrast to the usual value of 30-40 d. Unlike normal SNe Ia, the I light curve does not exhibit a secondary maximum, and the R light curve shows no sign of a plateau. Despite having prominent Si II absorption near 6150 A, characteristic of SNe Ia, SN 1991bg was spectroscopically peculiar at maximum brightness; e.g. a deep trough was present around 4200 A. Spectral synthesis suggests that much of this feature was produced by Ti II. Indeed, Ti II was present elsewhere in the spectrum whereas Fe II was weak or absent. Lines due to other intermediate-mass elements were also strong near maximum brightness, but the expansion velocity (~10 000 km s^-1^) was slightly lower than average (11 000-13 000 km s^-1^) for SNe Ia. A spectrum obtained 3 weeks past maximum has a rather narrow absorption line attributed to Na I D; the strength of this feature grew with time. In certain other respects the spectrum 3 weeks past maximum more closely resembles spectra of SNe Ib/Ic than of SNe Ia. We identify emission lines of [Ca II] λλ7291, 7324, O I λ7773, the Ca II infrared triplet, and (tentatively) [O I] λ5577, all superimposed on a complex blend of P Cygni scattering lines; apparently the nebular phase began very early in SN 1991bg. Three months past maximum the [Ca II] blend started to dominate the spectrum. The properties of SN 1991bg suggest that this object was produced by the slow deflagration of a white dwarf. The observed expansion velocity, however, is too large unless a sub-Chandrasekhar mass of material was ejected. An unusually low mass for the ejecta expanding at a nearly normal velocity, is consistent with the early transition to the nebular phase as well as with the rapid decline and the early knee in the V light curve. Thus, SN 1991bg may have resulted from the deflagration of only the outer layer of material on top of a C-O core, thereby leaving a compact remnant; alternatively, perhaps the complete explosion of a low-mass (~1.0 M_sun_) white dwarf was responsible. We show that some properties of the peculiar SN 1986G in Cen A were similar to those of SN 1991bg, and it may have been intrinsically dim by several magnitudes. Moreover, SN 1971I in NGC 5055 is known to have resembled SN 1986G; thus, SN 1991bg was probably a close cousin of these two objects. SN 1991bg clearly demonstrates that at least some SNe 1a are mavericks, both photometrically and spectroscopically. These objects must be carefully eliminated from samples of SNe 1a used in determinations of cosmological parameters.
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