The partial spontaneous breaking of rigid N=2 supersymmetry implies the existence of a massless N=1 Goldstone multiplet. In this paper we show that the spin-(1/2,1) Maxwell multiplet can play this role. We construct its full nonlinear transformation law and find the invariant Goldstone action. The spin-1 piece of the action turns out to be of Born-Infeld type, and the full superfield action is duality invariant. This leads us to conclude that the Goldstone multiplet can be associated with a D-brane solution of superstring theory for p=3. In addition, we find that N=1 chirality is preserved in the presence of the Goldstone-Maxwell multiplet. This allows us to couple it to N=1 chiral and gauge field multiplets. We find that arbitrary Kahler and superpotentials are consistent with partially broken N=2 supersymmetry.