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Data from two flights of a new superconducting magnetic spectrometer are reported. This instrument was capable of a direct matter-antimatter separation in the cosmic rays. Antimatter events would appear in the spectrometer as trajectories which curve in the opposite direction to common matter, because of their negative charge. A brief description of the equipment and of the characteristics of the instrument is presented, along with the data processing techniques used. A new upper limit on the amount of antimatter in primary cosmic rays has been established. The limits are considerably lower than those for any previous experiment.