Solar cells based on organic-inorganic halide perovskites have recently shown rapidly rising power conversion efficiencies, but exhibit unusual behaviour such as current-voltage hysteresis and a low-frequency giant dielectric response. Ionic transport has been suggested to be an important factor contributing to these effects; however, the chemical origin of this transport and the mobile species are unclear. Here, the activation energies for ionic migration in methylammonium lead iodide (CH3NH3PbI3) are derived from first principles, and are compared with kinetic data extracted from the current-voltage response of a perovskite-based solar cell. We identify the microscopic transport mechanisms, and find facile vacancy-assisted migration of iodide ions with an activation energy of 0.6 eV, in good agreement with the kinetic measurements. The results of this combined computational and experimental study suggest that hybrid halide perovskites are mixed ionic-electronic conductors, a finding that has major implications for solar cell device architectures.