Rifts are segmented linear depressions that are filled with sedimentary and igneous rocks; they form by extension and often evolve into plate boundaries. Flood basalts, a class of large igneous provinces (LIPs), are broad regions of extensive volcanism formed by sublithospheric processes. Typical rifts are not filled with flood basalts, and typical flood basalts are not associated with significant crustal extension and faulting. North America’s Midcontinent Rift (MCR) is an unusual combination, because its 3000-km length formed during a continental breakup event 1.1 Ga, but it contains an enormous volume of igneous rocks that are mostly flood basalt. We show that MCR volcanic rocks are significantly thicker than other flood basalts, due to their deposition in a narrow rift rather than across a broad region, giving the MCR a rift’s geometry but a LIP’s magma volume. Structural modeling of seismic-reflection data shows that LIP volcanics were deposited during two phases—an initial rift phase where flood basalts filled a fault-controlled extending basin and a postrift phase where LIP volcanics and sediments were deposited in a thermally subsiding sag basin without associated faulting. The crust thinned during the initial rifting phase and then rethickened during the postrift phase and later compression, yielding the present thicker crust observed seismologically. The restriction of extension to a single normal fault in each rift segment, steeply inward-dipping rift shoulders with sharp hinges, and persistence of volcanism after rifting ended gave rise to a deep flood basalt–filled rift geometry not observed in other presently active or ancient rifts. The unusual coincidence of a rift and LIP arose when a new rift associated with continental breakup interacted with a mantle plume or overrode anomalously hot or fertile upper mantle.