The measurement of the atmospheric muon spectrum is currently of great interest because of the study of atmospheric neutrinos and the claim of neutrino oscillations made in 1998 by the Super-Kamiokande Collaboration. A measurement of the muon flux is an indirect measure of the neutrino flux. Therefore, it can be used to improve the calculation of the atmospheric neutrino flux, which in turn can be compared with the observed neutrino rates in underground detectors. This article reports a new measurement of the μ+ and μ- spectra at several atmospheric depths in the momentum ranges 0.3 20 GeV/c and 0.3 40 GeV/c, respectively. The data were collected by the balloon-borne experiment CAPRICE98 during the ascent of the payload on 28 May 1998 from Fort Sumner, N. M. The experiment used the NMSU-WIZARD/CAPRICE 98 balloon-borne magnet spectrometer equipped with a gas ring imaging Cherenkov detector and a silicon-tungsten calorimeter.