An exclusive measurement of the Coulomb breakup of 8B into 7Be+p at 254A MeV allowed the study of the angular correlations of the breakup particles. These correlations demonstrate clearly that E1 multipolarity dominates and that E2 multipolarity can be neglected. By using a simple single-particle model for 8B and treating the breakup in first-order perturbation theory, we extract a zero-energy S factor of S17(0)=18.6+/-1.2+/-1.0 eV b, where the first error is experimental and the second one reflects the theoretical uncertainty in the extrapolation.