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Crews of high-altitude aircraft are exposed to radiation from galactic cosmic rays (GCRs). To help determine such exposures, the Atmospheric Ionizing Radiation Project, an international collaboration of 15 laboratories, made simultaneous radiation measurements with 14 instruments on a NASA ER-2 high-altitude airplane. The primary instrument was a sensitive extended-energy multisphere neutron spectrometer. Its detector responses were calculated for energies up to 100 GeV using the radiation transport code MCNPX 2.5.d with improved nuclear models and including the effects of the airplane structure. New calculations of GCR-induced particle spectra in the atmosphere were used to correct for spectrometer counts produced by protons, pions and light nuclear ions. Neutron spectra were unfolded from the corrected measured count rates using the deconvolution code MAXED 3.1. The results for the measured cosmic-ray neutron spectrum (thermal to >10 GeV), total neutron fluence rate, and neutron dose equivalent and effective dose rates, and their dependence on altitude and geomagnetic cut-off agree well with results from recent calculations of GCR-induced neutron spectra.