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 +/- 0.1)%, where the first uncertainty is statistical
and the second is systematic.  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.