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Measurements of the cosmic-ray hydrogen and helium spectra at energies from 20 to 800 TeV are presented. The experiments were performed on a series of twelve balloon flights, including several long duration Australia to South America and Antarctic circumpolar flights. No clear evidence is seen for a spectral break. Both the hydrogen and the helium spectra are consistent with power laws over the entire energy range, with integral spectral indices 1.80±0.04 and 1.68 for the protons and helium, respectively. The results are fully consistent with expectations based on supernova shock acceleration coupled with a" leaky box" model of propagation through the Galaxy.