The occurrence of earthquakes in the lower crust near continental rifts has long been puzzling, as the lower crust is generally thought to be too hot for brittle failure to occur. Such anomalous events have usually been explained in terms of the lower crust being cooler than normal. But if the lower crust is indeed cold enough to produce earthquakes, then the uppermost mantle beneath it should also be cold enough, and yet uppermost mantle earthquakes are not observed. Numerous lower-crustal earthquakes occur near the southwestern termination of the Taupo Volcanic Zone (TVZ), an active continental rift in New Zealand. Here we present three-dimensional tomographic imaging of seismic velocities and seismic attenuation in this region using data from a dense seismograph deployment. We find that crustal earthquakes accurately relocated with our three-dimensional seismic velocity model form a continuous band along the rift, deepening from mostly less than 10 km in the central TVZ to depths of 30-40 km in the lower crust, 30 km southwest of the termination of the volcanic zone. These earthquakes often occur in swarms, suggesting fluid movement in critically loaded fault zones. Seismic velocities within the band are also consistent with the presence of fluids, and the deepening seismicity parallels the boundary between high seismic attenuation (interpreted as partial melt) within the central TVZ and low seismic attenuation in the crust to the southwest. This linking of upper and lower-crustal seismicity and crustal structure allows us to propose a common explanation for all the seismicity, involving the weakening of faults on the periphery of an otherwise dry, mafic crust by hot fluids, including those exsolved from underlying melt. Such fluids may generally be an important driver of lower-crustal seismicity near continental rifts.