Superconductivity in FeSe has recently attracted a great deal of attention because it emerges out of an electronic nematic state of elusive character. Here we study both the electronic normal state and the superconducting gap structure using heat-capacity measurements on high-quality single crystals. The specific-heat curve, from 0.4 K to Tc = 9.1 K, is found to be consistent with a recent gap determination using Bogoliubov quasiparticle interference [P. O. Sprau et al., Science 357, 75 (2017)], however only if nodes are introduced on either the electron or the hole Fermi-surface sheets. Our analysis, which is consistent with quantum-oscillation measurements, relies on the presence of only two bands, and thus the fate of the theoretically predicted second electron pocket remains mysterious.