We present a measurement of the branching fraction and photon-energy spectrum for the decay B→Xsγ using data from the BABAR experiment. The data sample corresponds to an integrated luminosity of 210 fb-1, from which approximately 680 000 BB̅ events are tagged by a fully reconstructed hadronic decay of one of the B mesons. In the decay of the second B meson, an isolated high-energy photon is identified. We measure B(B→Xsγ)=(3.66±0.85stat±0.60syst)×10-4 for photon energies Eγ above 1.9 GeV in the B rest frame. From the measured spectrum we calculate the first and second moments for different minimum photon energies, which are used to extract the heavy-quark parameters mb and μπ2. In addition, measurements of the direct CP asymmetry and isospin asymmetry are presented.