The reactions gamma p --> K+ Lambda and gamma p --> K+ Sigma0 were measured in the energy range from threshold up to a photon energy of 2.6 GeV. The data were taken with the SAPHIR detector at the electron stretcher facility, ELSA. Results on cross sections and hyperon polarizations are presented as a function of kaon production angle and photon energy. The total cross section for Lambda production rises steeply with energy close to threshold, whereas the Sigma0 cross section rises slowly to a maximum at about E_gamma = 1.45 GeV. Cross sections together with their angular decompositions into Legendre polynomials suggest contributions from resonance production for both reactions. In general, the induced polarization of Lambda has negative values in the kaon forward direction and positive values in the backward direction. The magnitude varies with energy. The polarization of Sigma0 follows a similar angular and energy dependence as that of Lambda, but with opposite sign.