We present a new measurement of the cosmic-ray positron fraction at energies between 5 and 15 GeV with the balloon-borne HEAT-pbar instrument in the spring of 2000. The data presented here are compatible with our previous measurements, obtained with a different instrument. The combined data from the three HEAT flights indicate a small positron flux of nonstandard origin above 5 GeV. We compare the new measurement with earlier data obtained with the HEAT-e(+/-) instrument, during the opposite epoch of the solar cycle, and conclude that our measurements do not support predictions of charge sign dependent solar modulation of the positron abundance at 5 GeV.