We have studied the low-energy antiprotons in the cosmic rays by utilizing data obtained by the Balloon-borne Experiment with a Superconducting magnetic rigidity Spectrometer (BESS) flown in 1993 July from Lynn Lake, Manitoba, Canada. A detailed description of the event selection criteria and back-ground corrections is given. Seven antiprotons are found that give an antiproton flux of 6.4(-3.5)(+5.5) x 10(-3)(m(2) sr s GeV)(-1) and an antiproton/proton ratio of 5.2(-2.8)(+4.4) x 10(-6) in the 200-600 MeV energy range. These results are consistent with a secondary origin of low-energy cosmic-ray antiprotons within our uncertainties, but they still require the precise measurement of the spectrum shape below 500 MeV to clarify exactly the model of particle propagation and possible contributions from exotic sources.