With an integrated luminosity of 2.47 fb-1 recorded by the ATLAS experiment at the LHC, the exclusive decays Bs0→J/ψφ and Bd0→J/ψK∗0 of B mesons produced in pp collisions at √s=7 TeV are used to determine the ratio of fragmentation fractions fs/fd. From the observed Bs0→J/ψφ and Bd0→J/ψK∗0 yields, the quantity (fs/fd)[B(Bs0→J/ψφ)/B(Bd0→J/ψK∗0)] is measured to be 0.199±0.004(stat)±0.008(syst). Using a recent theory prediction for [B(Bs0→J/ψφ)/B(Bd0→J/ψK∗0)] yields (fs/fd)=0.240±0.004(stat)±0.010(syst)±0.017(th). This result is based on a new approach that provides a significant improvement of the world average.