  In a study of B+ --> J/psi gamma K+ decays, we find evidence for the
radiative decay X(3872) --> J/psi gamma with a statistical significance of 3.4
sigma. We measure the product of branching fractions BF(B+ --> X(3872)
K+).BF(X(3872) --> J/psi gamma) = (3.3 +/- 1.0 +/- 0.3) x 10^-6, where the
uncertainties are statistical and systematic, respectively. We also measure the
branching fraction BF(B+ --> chi_c1 K+) = (4.9 +/- 0.2 +/- 0.4) x 10^-4. These
results are obtained from (287+/-3) million BBar decays collected at the Y(4S)
resonance with the BaBar detector at the PEP-II B Factory at SLAC.
