Surface wave phase velocities from 29 earthquakes are used to map the shear velocity structure to ~350 km depth across the 950-km-long Rio Grande Rift Seismic Transect Experiment (LA RISTRA) seismic array in the southwest United States. Events from a range of back azimuths minimize the effects of multipathing. The resulting velocity model reveals a transition in lithospheric thickness from 200 km in the Great Plains to 45--55 km beneath the Rio Grande Rift, thickening beneath the Colorado Plateau to 120--150 km. The upper mantle low-velocity signature of the rift is roughly twice the width of its surface morphology. An asthenospheric low-velocity channel underlies the region west of the Great Plains and extends to 300 km depth. This channel is likely the result of warm mantle infill behind the sinking Farallon plate. Buoyant forces within this channel are sufficient to support much of the high elevation of the rift and plateau. No evidence for a deep mantle source is found beneath the rift, implying that present rifting is not driven by deep mantle upwelling. Velocities from 55 to 90 km beneath the rift axis are 10% slower than beneath the Great Plains, consistent with small amounts of partial melt. Low velocities extend to 200--300 km depth on either side of the rift but not directly beneath it, forming an inverted-U shape. This feature may reflect mantle that has cooled through vertical advection in a subadiabatic environment. Upwelling may be reinforced by small-scale convection caused by variations in lithospheric thickness and shallow mantle temperatures.