We observe the formation of shock waves in a Bose-Einstein condensate containing a large number of sodium atoms. The shock wave is initiated with a repulsive blue-detuned light barrier, intersecting the Bose-Einstein condensate, after which two shock fronts appear. We observe breaking of these waves when the size of these waves approaches the healing length of the condensate. At this time, the wave front splits into two parts and clear fringes appear. The experiment is modeled using an effective one-dimensional Gross-Pitaevskii-like equation and gives excellent quantitative agreement with the experiment, even though matter waves with wavelengths two orders of magnitude smaller than the healing length are present. In these experiments, no significant heating or particle loss is observed.