We report on a measurement of the branching fraction of the Cabibbo-suppressed decay Bu+→J/ψπ+, where J/ψ→μ+μ−. The data were collected by the Collider Detector at Fermilab during 1992–1995 and correspond to an integrated luminosity of 110pb−1 in p¯p collisions at s=1.8TeV. A signal of 28−9+10 events is observed and we determine the ratio of branching fractions B(Bu+→J/ψπ+)/B(Bu+→J/ψK+) to be [5.0−1.7+1.9(stat)±0.1(syst)]%. Using the world average value for B(Bu+→J/ψK+), we calculate the branching fraction B(Bu+→J/ψπ+) to be (5.0−1.9+2.1)×10−5. We also search for the decay Bc+→J/ψπ+ and report a 95% confidence level limit on σ(Bc+)B(Bc+→J/ψπ+)/σ(Bu+)B(Bu+→J/ψK+) as a function of the Bc+ lifetime.