We report cosmic-ray proton and helium spectra in energy ranges of 1-120 GeV nucleon-1 and 1-54 GeV nucleon-1, respectively, measured by a flight of the Balloon-borne Experiment with Superconducting Spectrometer (BESS) in 1998. The magnetic rigidity of the cosmic ray was reliably determined by highly precise measurement of the circular track in a uniform solenoidal magnetic field of 1 T. Those spectra were determined within overall uncertainties of ±5% for protons and ±10% for helium nuclei including statistical and systematic errors.