We report cosmic-ray proton and helium spectra in energy ranges of 1È120 GeV nucleon~1 and 1È54 GeV nucleon~1, respectively, measured by a Ñight 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 Ðeld of 1 T. Those spectra were determined within overall uncertainties of^5% for protons and^10% for helium nuclei including statistical and systematic errors.