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We investigate the generation of magnetic field by the Biermann battery in cosmological ionization fronts, using new simulations of the reionization of the universe by stars in protogalaxies. Two mechanisms are primarily responsible for magnetogenesis: i) the breakout of I-fronts from protogalaxies, and ii) the propagation of I-fronts through the high density neutral filaments which are part of the cosmic web. The first mechanism is dominant prior to overlapping of ionized regions (z ~ 7), whereas the second continues to operate even after that epoch. However, after overlap the field strength increase is largely due to the gas compression occurring as cosmic structures form. As a consequence, the magnetic field at z ~ 5 closely traces the gas density, and it is highly ordered on megaparsec scales. The mean mass-weighted field strength is B_0 ~ 10^{-19} G in the simulation box. There is a relatively well-defined, nearly linear correlation between B_0 and the baryonic mass of virialized objects, with B_0 ~ 10^{-18} G in the most massive objects (M ~ 10^9 M_sun) in our simulations. This is a lower limit, as lack of numerical resolution prevents us from following small scale dynamical processes which could amplify the field in protogalaxies. Although the field strengths we compute are probably adequate as seed fields for a galactic dynamo, the field is too small to have had significant effects on galaxy formation, on thermal conduction, or on cosmic ray transport in the intergalactic medium. It could, however, be observed in the intergalactic medium through innovative methods based on analysis of gamma-ray burst photon arrival times.