The role of the band structure in the Verwey transition in magnetite (Fe₃O₄) has been analysed within the framework of an exactly solvable two-band model based on an effective interionic Coulomb potential splitting the 3d band of the t_2g electrons. The model predicts an instability of the Verwey order below a certain ratio β/W of the Coulomb interaction parameter β and the bandwidth W. This instability can be understood in terms of a clear physical picture showing the impossibility of constructing a self-consistent ordered charge distribution in this case. A resemblance to the Cullen-Callen criterion is evident. It is shown as well that the influence of the band structure on the Verwey temperature vanishes rapidly when β/W increase beyond its critical value. Band-structure effects do provide an explanation, however, for the discontinuity in the Verwey temperature as a function of the concentration of cation dopants or the oxygen stoichiometry, which marks the transition from a first- to a second-order Verwey transition. In this respect, the model reproduces the experimental data quantitatively. Fits obtained by application of the model yield values of 0.037-0.04 eV for the Coulomb gap and 0.012-0.014 eV for the bandwidth. The obtained values of the bandwidth are typical for a strongly localized electron system and support a polaronic band picture.