We report on a measurement of the branching fraction of the
Cabibbo-suppressed decay B^+_u -> J/psi pi^+, where J/psi -> mu^+ mu^-.  The
data were collected by the Collider Detector at Fermilab during 1992-1995 and
correspond to an integrated luminosity of 110 pb^-1 in pbar p collisions at
sqrt s = 1.8 TeV.  A signal of 28^10_ -9 events is observed, and we determine
the ratio of branching fractions B(B^+_u-> J/psi pi^+)/B(B^+_u -> J/psi K^+)
to be (5.0^+1.9_ -1.7(stat.) +/- 0.1(syst.))%.  Using the world average value
for B(B^+_u -> J/psi K^+), we calculate the branching fraction B(B^+_u ->
J/psi pi^+) to be (5.0^+2.1_ -1.9) x 10^-5.  We also search for the decay
B^+_c -> J/psi pi^+ and report a 95% confidence level limit on sigma(B^+_c) .
B(B^+_c -> J/psi pi^+)/sigma(B^+_u) . B(B^+_u -> J/psi K^+) as a function of
the B^+_c lifetime.