From an exposure of 25.5~kiloton-years of the Super-Kamiokande detector, 900 muon-like and 983 electron-like single-ring atmospheric neutrino interactions were detected with momentum $p_e > 100$ MeV/$c$, $p_mu > 200$ MeV/$c$, and with visible energy less than 1.33 GeV. Using a detailed Monte Carlo simulation, the ratio $(mu/e)_{DATA}/(mu/e)_{MC}$ was measured to be $0.61 pm 0.03(stat.) pm 0.05(sys.)$, consistent with previous results from the Kamiokande, IMB and Soudan-2 experiments, and smaller than expected from theoretical models of atmospheric neutrino production.