We report the first infrared and optical measurements of the absolute conductivity of FeSi. Above 200 K the ac conductivity of FeSi resembles that of a dirty metal, while at low temperature most of the conductivity below 60 meV is depleted, consistent with an energy gap of that magnitude. Thus in contrast to what is expected from a picture based on simple thermal activation, the gap disappears at a temperature which is low relative to its size. Moreover, the spectral weight lost below the gap does not reappear just above the gap, but is instead distributed over an energy range of order eV.