We report the first evidence for a fully reconstructed decay mode of the B(c)+- meson in the channel B(c)+- --> J/psi pi+-, with J/psi --> mu+ mu-. The analysis is based on an integrated luminosity of 360 pb(-1) in pp collisions at 1.96 TeV center of mass energy collected by the Collider Detector at Fermilab. We observe 14.6 +/- 4.6 signal events with a background of 7.1 +/- 0.9 events, and a fit to the J/psi pi+-mass spectrum yields a B(c)+- mass of 6285.7 +/- 5.3(stat) +/- 1.2(syst) MeV/c2. The probability of a peak of this magnitude occurring by random fluctuation in the search region is estimated as 0.012%.