Final results from an exclusive measurement of the Coulomb breakup of 8B into 7Be+p at 254 A MeV are reported. Energy-differential Coulomb-breakup cross sections are analyzed using a potential model of 8B and first-order perturbation theory. The deduced astrophysical S_17 factors are in good agreement with the most recent direct 7Be(p,gamma)8B measurements and follow closely the energy dependence predicted by the cluster-model description of 8B by Descouvemont. We extract a zero-energy S_17 factor of 20.6 +- 0.8 (stat) +- 1.2 (syst) eV b.