We report on the absolute antiproton flux and the antiproton to proton ratio in the energy range 0.62-3.19 GeV at the top of the atmosphere, measured by the balloon-borne experiment CAPRICE flown from Lynn Lake, Manitoba, Canada, on 1994 August 8-9. The experiment used the New Mexico State University WiZard/CAPRICE balloon-borne magnet spectrometer equipped with a solid radiator Ring Imaging Cherenkov (RICH) detector and a silicon-tungsten calorimeter for particle identification. This is the first time a RICH is used together with an imaging calorimeter in a balloon experiment, and it allows antiprotons to be clearly identified over the rigidity range 1.2-4 GV. Nine antiprotons were identified in the energy range 0.62-3.19 GeV at the top of the atmosphere. The data were collected over 18 hr at a mean residual atmosphere of 3.9 g cm-2. The absolute antiproton flux is consistent with a pure secondary production of antiprotons during the propagation of cosmic rays in the Galaxy.