We report a measurement of the ratio of branching fractions of the decays B±→J/ψπ± and B±→J/ψK± using the CDF II detector at the Fermilab Tevatron Collider. The signal from the Cabbibo-suppressed B±→J/ψπ± decay is separated from B±→J/ψK± using the B±→J/ψK± invariant mass distribution and the kinematical differences of the hadron track in the two decay modes. From a sample of 220  pb−1 of pp collisions at s√=1.96  TeV, we observe 91±15 B±→J/ψπ± events together with 1883±34 B±→J/ψK± events. The ratio of branching fractions is found to be B(B±→J/ψπ±)/B(B±→J/ψK±)=(4.86±0.82(stat)±0.15(syst))%.