We report the results of a search for a charmoniumlike state produced in the process gamma gamma -> omega J/psi in the 3.9-4.2 GeV = c(2) mass region. We observe a significant enhancement, which is well described by a resonant shape with mass M = (3915 +/- 3 +/- 2_ MeV/c(2) and total width Gamma = (17 +/- 10 +/- 3) MeV. This enhancement may be related to one or more of the three charmoniumlike states so far reported in the 3.90-3.95 GeV= c(2) mass region.