The high elevations of the Himalaya and Tibet result from the continuing collision between India and Asia, which started more than 60 million years ago(1-4). From geological and seismic studies of the slip rate of faults in Asia(5), it is believed that approximately one-third of the present-day convergence rate between India and Asia (58 +/- 4mm yr(-1)) is responsible for the shortening, uplift and moderate seismicity of the Himalaya. Great earthquakes also occur infrequently in this region, releasing in minutes the elastic strain accumulated near the boundary zone over several centuries, and accounting for most of the advance of the Himalaya over the plains of India. The recurrence time for these great earthquakes is determined by the rate of slip of India beneath Tibet, which has hitherto been estimated indirectly from global plate motions(6), from the slip rates of faults in Asia(7,8), from seismic productivity(9), and from the advance of sediments on the northern Ganges plain(10). Here we report geodetic measurements, using the Global Positioning System (GPS), of the rate of contraction across the Himalaya, which we find to be 17.52 +/- 2 mm yr(-1). From the form of the deformation field, we estimate the rate of slip of India beneath Tibet to be 20.5 +/- 2 mm yr(-1). Strain sufficient to drive one or more great Himalayan earthquakes, with slip similar to that accompanying the magnitude 8.1 Bihar/Nepal 1934 earthquake, may currently be available in western Nepal.