The energy spectra of cosmic-ray protons and helium near solar minimum were precisely measured with BESS-Polar II (Balloon-borne Experiment with a Superconducting Spectrometer) during a long-duration flight over Antarctica in December 2007 and January 2008, and are discussed here. The absolute fluxes and spectral shapes of primary protons and helium probe the origin and the propagation history of cosmic rays in the Galaxy. The spectra are also essential as inputs to calculate the spectrum of cosmic-ray antiprotons, which are secondary products of cosmic-ray interactions with the interstellar gas. To optimize the measurement of the magnetic rigidity of incident particles, obtained from the curvature of their trajectories in a solenoidal magnetic field of 0.8 Tesla, an improved calibration of the central JET-type drift chamber and two inner drift chambers was developed. Using this, we obtained absolute spectra of primary cosmic-ray protons up to around 120 GeV and helium up to around 50 GeV/nucleon.