The Gamburtsev Subglacial Mountains are the least understood tectonic feature on Earth, because they are completely hidden beneath the East Antarctic Ice Sheet. Their high elevation and youthful Alpine topography, combined with their location on the East Antarctic craton, creates a paradox that has puzzled researchers since the mountains were discovered in 1958. The preservation of Alpine topography in the Gamburtsevs may reflect extremely low long-term erosion rates beneath the ice sheet, but the mountains’ origin remains problematic. Here we present the first comprehensive view of the crustal architecture and uplift mechanisms for the Gamburtsevs, derived from radar, gravity and magnetic data. The geophysical data define a 2,500-km-long rift system in East Antarctica surrounding the Gamburtsevs, and a thick crustal root beneath the range. We propose that the root formed during the Proterozoic assembly of interior East Antarctica (possibly about 1 Gyr ago), was preserved as in some old orogens and was rejuvenated during much later Permian (roughly 250 Myr ago) and Cretaceous (roughly 100 Myr ago) rifting. Much like East Africa, the interior of East Antarctica is a mosaic of Precambrian provinces affected by rifting processes. Our models show that the combination of rift-flank uplift, root buoyancy and the isostatic response to fluvial and glacial erosion explains the high elevation and relief of the Gamburtsevs. The evolution of the Gamburtsevs demonstrates that rifting and preserved orogenic roots can produce broad regions of high topography in continental interiors without significantly modifying the underlying Precambrian lithosphere.