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We present the design of the gamma-ray and neutron spectrometer (GR/NS) for Dawn, which is a NASA Discovery-class mission to explore two of the largest main-belt asteroids, 1 Ceres and 4 Vesta, whose accretion is believed to have been interrupted by the early formation of Jupiter. Dawn will determine the composition and structure of these protoplanetary bodies, providing context for a large number of primitive meteorites in our sample collection and a better understanding of processes occurring shortly after the onset of condensation of the solar nebula. The Dawn GR/NS design draws on experience from the successful Lunar Prospector and Mars Odyssey missions to enable accurate mapping of the surface composition and stratigraphy of major elements, radioactive elements, and hydrogen at both asteroids. Here, we describe the overall design of the GR/NS and compare the expected performance of the neutron spectrometer subsystem to the neutron spectrometer on Mars Odyssey. We also describe radiation damage studies carried out on CdZnTe detectors, which will be components of the primary gamma-ray spectrometer on Dawn. We conclude that provisions for annealing at moderate temperatures (40°C to 60°C) must be made to ensure that the spectrometer will function optimally over the nine-year mission.