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We report the first evidence for a fully reconstructed decay mode of the Bc± meson in the channel Bc± → J/ψπ±, with J/ψ → μ+ μ-. The analysis is based on an integrated luminosity of 360 pb-1 in pp̅ 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 Bc± 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%.