We report on experimental investigations of the propagation of cracks in the brittle plastic polymethyl methacrylate (PMMA). Velocity measurements with resolution an order of magnitude better than previous experiments reveal the existence of a critical velocity (330\ifmmode\pm\else\textpm\fi{}30 m/s) at which the velocity of the crack tip begins to oscillate, the dynamics of the crack abruptly change, and a periodic pattern is formed on the crack surface. Beyond the critical point the amplitude of the oscillations depends linearly on the mean velocity of the crack. The existence of this instability may explain the failure of theoretical predictions of crack dynamics and provides a mechanism for the enhanced dissipation observed experimentally in the fracture of brittle materials.