We report a detailed study, using neutron scattering, transport and magnetization measurements, of the interplay between superconducting (SC) and spin density wave (SDW) order in La2CuO4+y. Both kinds of order set in below the same critical temperature. However, the SDW order grows with applied magnetic field, whereas SC order is suppressed. Most importantly, the field dependence of the SDW Bragg peak intensity has a cusp at zero field, as predicted by a recent theory of competing SDW and SC order. This leads us to conclude that there is a repulsive coupling between the two order parameters. The question of whether the two kinds of order coexist or microscopically phase separate is discussed.