Antiparticles account for a small fraction of cosmic rays and are known to be produced in interactions between cosmic-ray nuclei and atoms in the interstellar medium, which is referred to as a 'secondary source'. Positrons might also originate in objects such as pulsars and microquasars or through dark matter annihilation, which would be 'primary sources'. Previous statistically limited measurements of the ratio of positron and electron fluxes have been interpreted as evidence for a primary source for the positrons, as has an increase in the total electron+positron flux at energies between 300 and 600 GeV (ref. 8). Here we report a measurement of the positron fraction in the energy range 1.5-100 GeV. We find that the positron fraction increases sharply over much of that range, in a way that appears to be completely inconsistent with secondary sources. We therefore conclude that a primary source, be it an astrophysical object or dark matter annihilation, is necessary.