Long known for its peculiar resistivity, showing a thus far unexplained anomalous peak as a function of temperature, ZrTe5 has recently received rising attention in a somewhat different context. While both theoretical and experimental results seem to point to a nontrivial topology of the low-energy electronic states, there is no agreement on the nature of their topological character. Here, by an angle-resolved photoemission study of the evolution of the band structure with temperature and surface doping, we show that (i) the material presents a van Hove singularity close to the Fermi level, and (ii) no surface states exist at the (010) surface. These findings reconcile band structure measurements with transport results and establish the topology of this puzzling compound.