The global mid-ocean ridge system, where tectonic plates diverge, is traditionally thought of as the largest single volcanic feature on the Earth. Yet, wide expanses of smooth sea floor in the easternmost part of the Southwest Indian Ridge in the Indian Ocean lacks the hummocky morphology that is typical for submarine volcanism. At other slow-spreading ridges, the sea floor can extend by faulting the existing lithosphere, along only one side of the ridge axis. However, the smooth sea floor in the easternmost Southwest Indian Ridge also lacks the corrugated texture created by such faulting. Instead, the sea floor is smooth on both sides of the ridge axis and is thought to be composed of altered mantle-derived rocks. Here we use side-scan sonar to image the sea floor and dredge samples to analyse the composition of two sections of the Southwest Indian Ridge, between 62 degrees 05'E and 64 degrees 40'E, where the sea floor formed over the past 11 million years. We show that the smooth floor is almost entirely composed of seawater-altered mantle-derived rocks that were brought to the surface by large detachment faults on both sides of the ridge axis. Faulting accommodates almost 100% of plate divergence and the detachment faults have repeatedly flipped polarity. We suggest that this tectonic process could also explain the exhumation of mantle-derived rocks at the magma-poor margins of rifted continents.