An experiment (E166) at the Stanford Linear Accelerator Center has demonstrated a scheme in which a multi-GeV electron beam passed through a helical undulator to generate multi-MeV, circularly polarized photons which were then converted in a thin target to produce positrons (and electrons) with longitudinal polarization above 80% at 6 MeV. The results are in agreement with GEANT4 simulations that include the dominant polarization-dependent interactions of electrons, positrons, and photons in matter.