We report on the discovery and analysis of a striking neutral sodium gas tail associated with comet C/1995 O1 Hale-Bopp. Sodium D line emission has been observed at heliocentric distance r<=1.4 AU in some long-period comets, and the presence of neutral sodium in the tailward direction of a few bright comets has been noted, but the extent, and in particular the source, has never been clear. Here we describe the first observations and analysis of a neutral sodium gas tail in comet Hale-Bopp, which is entirely different from the previously known ion and dust tails. We show that the observed characteristics of this third type of tail are consistent with it being produced by radiation pressure due to resonance fluorescence of sodium atoms and that the lifetime for photoionization is consistent with recent theoretical calculations.