Optical and photoemission experiments reveal unexpected spectral signatures of one-dimensional band insulators. In the model compound (NbSe (4))3I the optical conductivity decays as a power law sigma(1)(omega) approximately omega(-4.25) above a sharp gap edge. Photoemission observes both the valence and a shadow band, produced by a commensurate superstructure. We identify an optical and photoemission band gap consistent with other measurements but much smaller than the energy scale defined by the dispersion of the band peak in the photoemission spectra.