We present a study of the decays B0,+→J/ψωK0,+ using 383×106BBˉ events obtained with the \babar detector at PEP-II. We observe Y(3940)→J/ψω, with mass 3914.6−3.4+3.8(stat)±2.0(syst) \mevcc, and width 34−8+12(stat)±5(syst) \mev. The ratio of B0 and B+ decay to YK is 0.27−0.23+0.28(stat)−0.01+0.04(syst), and the relevant B0 and B+ branching fractions are reported.