There is strong experimental evidence that the superconductor Sr₂RuO₄ has a chiral p-wave order parameter. This symmetry does not require that the associated gap has nodes, yet specific heat, ultrasound, and thermal conductivity measurements indicate the presence of nodes in the superconducting gap structure of Sr₂RuO₄. Theoretical scenarios have been proposed to account for the existence of deep minima or accidental nodes (minima tuned to zero or below by material parameters) within a p-wave state. Other scenarios propose chiral d-wave and f-wave states, with horizontal and vertical line nodes, respectively. To elucidate the nodal structure of the gap, it is essential to know whether the lines of nodes (or minima) are vertical (parallel to the tetragonal c axis) or horizontal (perpendicular to the c axis). Here, we report thermal conductivity measurements on single crystals of Sr₂RuO₄ down to 50 mK for currents parallel and perpendicular to the c axis. We find that there is substantial quasiparticle transport in the T = 0 limit for both current directions. A magnetic field H immediately excites quasiparticles with velocities both in the basal plane and in the c direction. Our data down to T_c/30 and down to H_c2=100 show no evidence that the nodes are in fact deep minima. Relative to the normal state, the thermal conductivity of the superconducting state is found to be very similar for the two current directions, from H = 0 to H = H_c2. These findings show that the gap structure of Sr₂RuO₄ consists of vertical line nodes. This rules out a chiral d-wave state. Given that the c-axis dispersion (warping) of the Fermi surface in Sr₂RuO₄ varies strongly from sheet to sheet, the small a - c anisotropy suggests that the line nodes are present on all three sheets of the Fermi surface. If imposed by symmetry, vertical line nodes would be inconsistent with a p-wave order parameter for Sr₂RuO₄. To reconcile the gap structure revealed by our data with a p-wave state, a mechanism must be found that produces accidental line nodes in Sr₂RuO₄.