Cosmic rays are primarily protons and other nuclei, but there is a small flux of electrons and an even smaller flux of positrons. Cosmic-ray positrons can be produced by astrophysical accelerators, by collisions of cosmic-ray protons with interstellar gas, or by dark matter annihilation. We used the Fermi Gamma-ray Space Telescope to measure the charge-separated electron and positron energy spectra. Because Fermi does not have an onboard magnet, we used the Earth's magnetic field to distinguish positrons and electrons. We confirmed the PAMELA discovery that the positron fraction is rising with energy between 10 and 100 GeV and measured the positron flux for the first time in the 100-200 GeV range. Explaining the positron excess remains an outstanding question.