Evidence is reported for a narrow structure near the J/psivarphi threshold in exclusive B;{+} --> J/psivarphiK;{+} decays produced in p[over]p collisions at sqrt[s] = 1.96 TeV. A signal of 14 +/- 5 events, with statistical significance in excess of 3.8 standard deviations, is observed in a data sample corresponding to an integrated luminosity of 2.7 fb;{-1}, collected by the CDF II detector. The mass and natural width of the structure are measured to be 4143.0 +/- 2.9(stat) +/- 1.2(syst) MeV/c;{2} and 11.7_{-5.0};{+8.3}(stat) +/- 3.7(syst) MeV/c;{2}.