We present new BATSE Earth occultation observations of the 25 keV-1.8 MeV spectrum and variability of Cyg X-1 made between 1993 August and 1994 May. We observed that the normal soft γ-ray spectrum (γ2) of Cyg X-1 has two components: a Comptonized part seen below 300 keV and a high-energy tail in the 0.3-2 MeV range that was only hinted at in previous γ2 spectrum observed by HEAO 3. The source went through an extended sequence of changes between 1993 August and 1994 May; the 45-140 keV flux first decreased steadily from ~γ2 to below the γ1 flux level seen previously by HEAO 3 in 1979 to a new level, γ0, roughly one-quarter of its intensity over a period of ~140 days. The flux remained at this low level for about 40 days before returning swiftly (~20 days) to approximately the initial γ2 level. The γ2 spectrum may be interpreted in terms of an interacting two-region model, consisting of a high-temperature (~210-250 keV) core embedded in an ~50 keV corona. In this scenario, the observed 25-300 keV photons were produced by Compton scattering of soft photons (~0.5 keV) by the hot electrons in the outer corona. These same hard X-rays were further upscattered by a population of energetic electrons in the inner core, producing the spectral tail above 300 keV. During the excursion of the 45-140 keV flux from the γ2 to the γ0 level, the spectrum evolved to a form consistent with either a power law with a photon index of ~2.6 or a single-temperature Compton model with an electron temperature, kT, of 108 +/- 11 keV and an optical depth, τ, of 0.40 +/- 0.06 and then returned essentially to the original γ2 spectrum at the end of the active period. The overall cooling of the system during the low-flux period may be due to an increase in the soft photon population that effectively quenched the hot electrons in these regions through Compton scattering.