We report the first evidence for a fully reconstructed decay mode of the B±c meson in the channel B±c→J/ψπ±, with J/ψ→μ+μ−. 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/ψπ± 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%.