The most spectacular example of a plate convergence event on Earth is the motion of the Indian plate towards Eurasia at speeds in excess of 18 cm yr(-1) (ref. 1), and the subsequent collision. Continental buoyancy usually stalls subduction shortly after collision, as is seen in most sections of the Alpine-Himalayan chain. However, in the Indian section of this chain, plate velocities were merely reduced by a factor of about three when the Indian continental margin impinged on the Eurasian trench about 50 million years ago. Plate convergence, accompanied by Eurasian indentation, persisted throughout the Cenozoic era(1-3), suggesting that the driving forces of convergence did not vanish on continental collision. Here we estimate the density of the Greater Indian continent, after its upper crust is scraped off at the Himalayan front, and find that the continental plate is readily subductable. Using numerical models, we show that subduction of such a dense continent reduces convergence by a factor similar to that observed. In addition, an imbalance between ridge push and slab pull can develop and cause trench advance and indentation. We conclude that the subduction of the dense Indian continental slab provides a significant driving force for the current India-Asia convergence and explains the documented evolution of plate velocities following continental collision.