Soft gamma repeaters (SGRs) are a class of rare, high-energy galactic transients that have episodes of short (similar to 1 s), soft (similar to 30 keV), intense (similar to 100 Crab), gamma-ray bursts. We report an analysis of the X-ray emission from 95 SGR 1806-20 events observed by the international Cometary Explorer. The spectral shape remains remarkably constant for bursts that differ in intensity by a range of 50. Below similar to 15 keV the number spectrum falls off rapidly such that we can estimate the total intensity of the events. Assuming that SGR 1806-20 is associated with the supernova remnant G10.0-0.3 (Kulkarni and Frail 1993; Murakami et al. 1994), the brightest events had a total luminosity of 1.8 x 10(42) ergs s(-1), a factor of 2 x 10(4) above the Eddington limit. A third of the emission was above 30 keV. There are at least three processes that are consistent with the spectral rollover below 15 keV. (1) The rollover is consistent with some forms of self-absorption. Typical thermal temperatures are similar to 20 keV and require an emitting surface with a radius between 10 and 50 km. The lack of spectral variability implies that only the size of the emitting surface varies from event to event. If the process is thermal synchrotron, the required magnetic field might be too small to confine the plasma against the super-Eddington flux. (2) The low-energy rollover could be due to photoelectric absorption by 10(24) hydrogen atoms cm(-2) of neutral material with a cosmic abundance. This assumes a continuum similar to thermal bremsstrahlung with a temperature of similar to 22 keV. The material is most likely to be associated with the object as circumstellar matter a few AU from the central source rather than foreground clouds or directly at the site of the energy release. (3) Emission in the two lowest harmonics from a 1.3 x 10(12) G field would appear as Doppler-broadened lines and fall off rapidly below 15 keV.