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An exclusive measurement of the Coulomb breakup of $^{8}\mathrm{B}$ into $^{7}\mathrm{Be}$$+p$ at 254A MeV was used to infer the low-energy $^{7}\mathrm{Be}$($p,\ensuremath{\gamma}$)$^{8}\mathrm{B}$ cross section. The radioactive $^{8}\mathrm{B}$ beam was produced by projectile fragmentation of 350A MeV $^{12}\mathrm{C}$ and separated with the FRagment Separator (FRS) at Gesellschaft f\"ur Schwerionenforschung in Darmstadt, Germany. The Coulomb-breakup products were momentum-analyzed in the KaoS magnetic spectrometer; particular emphasis was placed on the angular correlations of the breakup particles. These correlations demonstrate clearly that $E1$ multipolarity dominates within the angular cuts selected for the analysis. The deduced astrophysical ${S}_{17}$ factors exhibit good agreement with the most recent direct $^{7}\mathrm{Be}$($p,\ensuremath{\gamma}$)$^{8}\mathrm{B}$ measurements. By using the energy dependence of ${S}_{17}$ according to the recently refined cluster model for $^{8}\mathrm{B}$ of P. Descouvemont [Phys. Rev. C 70, 065802 (2004)], we extract a zero-energy S factor of ${S}_{17}(0)=20.6\ifmmode\pm\else\textpm\fi{}0.8(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}1.2(\mathrm{syst})$ eV b. These errors do not include the uncertainty of the theoretical model to extrapolate to zero relative energy, estimated by Descouvemont to be about 5%.