We present a new measurement of the cosmic-ray positron fraction e(+)/
(e(+) + e(-)) obtained from the first balloon flight of the High Energ
y Antimatter Telescope (HEAT). Using a magnet spectrometer combined wi
th a transition radiation detector, an electromagnetic calorimeter, an
d time-of-flight counters we have achieved a high degree of background
rejection. Our results do not indicate a major contribution to the po
sitron flux from primary sources. In particular, we see no evidence fo
r the significant rise in the positron fraction at energies above simi
lar to 10 GeV previously reported.