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A new technique to measure the ratio of b quark fragmentation fractions in pp̅ collisions is described. Using a 70-pb-1 sample of low-mass dimuon trigger data recorded with the Collider Detector at Fermilab, we identify B mesons by observing the double semileptonic decays b→cμX with c→sμ X. By counting the numbers of K*(892) 0, K*(892) 1, and Φ (1020) mesons produced in association with these muon pairs, we measure the ratio of strange to nonstrange B meson production to be fs /( fu+ i>fd)=[21.0±3.6(stat) -3.0+3.8 (syst)]%. This measurement is the most precise available from hadron collisions to date. Limits on the branching fractions of semileptonic charm meson decays with K1 (1270), K1* (1410), and K2* (1430) mesons in the final state are also obtained.