We present 1.20-18.5 μm infrared (IR) spectrophotometric measurements of comet C/1995 O1 (Hale-Bopp) during 1997 February 15 and 20 UT. The spectral energy distribution (SED) was dominated by scattering and thermal emission from submicron sized dust grains that were unusually small. Hale-Bopp's surprising brightness may have been largely a result of the properties of its coma grains rather than the size of its nucleus. The thermal emission continuum from the grains had a superheat of S=Tcolor/TBB ≈ 1.84, the peak of the 10 μm silicate emission feature was 1.81 mag above the carbon grain continuum, and the albedo (reflectivity) of the grains was ≈ 0.41 at a scattering angle of θ ≈ 144°. These are the highest values for these empirical parameters ever observed in 20 years of optical/IR measurements of bright comets. The observations indicate that the optically important grains dominating the visual scattering and near-IR emission from the coma had an average radius of a≤0.4 μm. The strong silicate feature is produced by grains with a similar size range. These dust radii are comparable to the radii of the grains that condense in the outflows of some novae ("stardust") but still about 10 times larger than the average radius of the grains that produce the general interstellar extinction.