The BESS-Polar Collaboration measured the energy spectra of cosmic-ray protons and helium during two long-duration balloon flights over Antarctica in 2004 December and 2007 December at substantially different levels of solar modulation. Proton and helium spectra probe the origin and propagation history of cosmic rays in the galaxy, and are essential to calculations of the expected spectra of cosmic-ray antiprotons, positrons, and electrons from interactions of primary cosmic-ray nuclei with the interstellar gas, and to calculations of atmospheric muons and neutrinos. We report absolute spectra at the top of the atmosphere for cosmic-ray protons in the kinetic energy range 0.2-160 GeV and helium nuclei in the range 0.15-80 GeV/nucleon. The corresponding magnetic-rigidity ranges are 0.6-160 GV for protons and 1.1-160 GV for helium. These spectra are compared to measurements from previous BESS flights and from ATIC-2, PAMELA, and AMS-02. We also report the ratio of the proton and helium fluxes from 1.1 to 160 GV and compare this to the ratios from PAMELA and AMS-02.