The orbits of binary stars precess as a result of general relativistic effects, forces arising from the asphericity of the stars, and forces from any additional stars or planets in the system. For most binaries, the theoretical and observed precession rates are in agreement. One system, however-DI Herculis-has resisted explanation for 30 years. The observed precession rate is a factor of four slower than the theoretical rate, a disagreement that once was interpreted as evidence for a failure of general relativity. Among the contemporary explanations are the existence of a circumbinary planet and a large tilt of the stellar spin axes with respect to the orbit. Here we report that both stars of DI Herculis rotate with their spin axes nearly perpendicular to the orbital axis (contrary to the usual assumption for close binary stars). The rotationally induced stellar oblateness causes precession in the direction opposite to that of relativistic precession, thereby reconciling the theoretical and observed rates.