Cosmic rays at the Earth include a secondary component originating in collisions of primary particles with the diffuse interstellar gas. The secondary cosmic rays are relatively rare but carry important information on the Galactic propagation of the primary particles. The secondary component includes a small fraction of antimatter particles, positrons and antiprotons. In addition, positrons and antiprotons may also come from unusual sources and possibly provide insight into new physics. For instance, the annihilation of heavy supersymmetric dark matter particles within the Galactic halo could lead to positrons or antiprotons with distinctive energy signatures. With the High-Energy Antimatter Telescope (HEAT) balloon-borne instrument, we have measured the abundances of positrons and electrons at energies between 1 and 50 GeV. The data suggest that indeed a small additional antimatter component may be present that cannot be explained by a purely secondary production mechanism. Here we describe the signature of the effect and discuss its possible origin.