As part of a series of experiments to search for antimatter in cosmic
rays, the New Mexico State University balloon-borne magnet spectromete
r was configured for a flight to study positrons. Two completely new i
nstruments, a transition radiation detector and a silicon-tungsten ima
ging calorimeter, were added to the magnet spectrometer. These two det
ectors provided a proton rejection factor better than 3 x 10(4). 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 repo
rt here the measured fraction of positrons e(+)/(e(+) + e(-)) from sim
ilar to 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 en
ergy, and our results are consistent with a constant fraction of 0.078
+/- 0.016 over the entire energy region.