As part of a series of experiments to search for antimatter in cosmic rays, the New Mexico State University balloon-borne magnet spectrometer was configured for a flight to study positrons. Two completely new instruments, a transition radiation detector and a silicon-tungsten imaging calorimeter, were added to the magnet spectrometer. These two detectors provided a proton rejection factor better than 3 x 104. This instrument was flown from Fort Sumner, New Mexico, at an average depth of 4.5 g cm-2 of residual atmosphere for a period of 25 hr. We report here the measured fraction of positrons e+/(e+ + e-) from ~5 to 60 GeV at the top of the atmosphere. Our measurements do not show any compelling evidence for an increase in this ratio with energy, and our results are consistent with a constant fraction of 0.078 +/- 0.016 over the entire energy region.