Two pulsars (PSR J0737-3039A and B) were recently discovered in highly relativistic orbits around one another. The system contains a rapidly rotating pulsar with a spin period of 22.7ms and a slow companion with a spin period of 2.77s, referred to here as `A' and `B', respectively. A unique property of the system is that the pulsed radio flux from B increases systematically by almost two orders of magnitude during two short portions of its orbit. Here we report a geometrical model of the system that simultaneously explains the intensity variations of B and provides constraints on the spin axis orientation and emission geometry of A. Our model assumes that B's pulsed radio flux increases when illuminated by emission from A. We predict that A's pulse profile will evolve considerably over the next several years owing to geodetic precession until it disappears entirely in 15-20 years.