This Letter presents the first measurement of event-by-event fluctuations of the elliptic flow parameter v(2) in Au+Au collisions at square root(s(NN))=200 GeV as a function of collision centrality. The relative nonstatistical fluctuations of the v(2) parameter are found to be approximately 40%. The results, including contributions from event-by-event elliptic flow fluctuations and from azimuthal correlations that are unrelated to the reaction plane (nonflow correlations), establish an upper limit on the magnitude of underlying elliptic flow fluctuations. This limit is consistent with predictions based on spatial fluctuations of the participating nucleons in the initial nuclear overlap region. These results provide important constraints on models of the initial state and hydrodynamic evolution of relativistic heavy ion collisions.