Abstract: A supersolid, a counterintuitive quantum state in which a rigid lattice of particles flows without resistance, has to date not been unambiguously realized. Here we reveal a supersolid ground state of excitons in a double-layer semiconductor heterostructure over a wide range of layer separations outside the focus of recent experiments. This supersolid conforms to the original Chester supersolid with one exciton per supersolid site, as distinct from the alternative version reported in cold-atom systems of a periodic density modulation or clustering of the superfluid. We provide the phase diagram augmented by the supersolid. This new phase appears at layer separations much smaller than the predicted exciton normal solid, and it persists up to a solid-solid transition where the quantum phase coherence collapses. The ranges of layer separations and exciton densities in our phase diagram are well within reach of the current experimental capabilities.