With an integrated luminosity of $2.47\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ recorded by the ATLAS experiment at the LHC, the exclusive decays ${B}_{s}^{0}\ensuremath{\rightarrow}J/\ensuremath{\psi}\ensuremath{\phi}$ and ${B}_{d}^{0}\ensuremath{\rightarrow}J/\ensuremath{\psi}{K}^{*0}$ of $B$ mesons produced in $pp$ collisions at $\sqrt{s}=7\text{ }\text{ }\mathrm{TeV}$ are used to determine the ratio of fragmentation fractions ${f}_{s}/{f}_{d}$. From the observed ${B}_{s}^{0}\ensuremath{\rightarrow}J/\ensuremath{\psi}\ensuremath{\phi}$ and ${B}_{d}^{0}\ensuremath{\rightarrow}J/\ensuremath{\psi}{K}^{*0}$ yields, the quantity $({f}_{s}/{f}_{d})[\mathcal{B}({B}_{s}^{0}\ensuremath{\rightarrow}J/\ensuremath{\psi}\ensuremath{\phi})/\mathcal{B}({B}_{d}^{0}\ensuremath{\rightarrow}J/\ensuremath{\psi}{K}^{*0})]$ is measured to be $0.199\ifmmode\pm\else\textpm\fi{}0.004(\text{stat})\ifmmode\pm\else\textpm\fi{}\phantom{\rule{0ex}{0ex}}0.008(\text{syst})$. Using a recent theory prediction for $[\mathcal{B}({B}_{s}^{0}\ensuremath{\rightarrow}J/\ensuremath{\psi}\ensuremath{\phi})/\mathcal{B}({B}_{d}^{0}\ensuremath{\rightarrow}J/\ensuremath{\psi}{K}^{*0})]$ yields $({f}_{s}/{f}_{d})=0.240\ifmmode\pm\else\textpm\fi{}0.004(\text{stat})\ifmmode\pm\else\textpm\fi{}0.010(\text{syst})\ifmmode\pm\else\textpm\fi{}0.017(\mathrm{th})$. This result is based on a new approach that provides a significant improvement of the world average.