We report measurements of B→χc1γK and χc2γK decays using 772×106 BB¯ events collected at the Υ(4S) resonance with the Belle detector at the KEKB asymmetric-energy e+e- collider. Evidence of a new resonance in the χc1γ final state is found with a statistical significance of 3.8σ. This state has a mass of 3823.1±1.8(stat)±0.7(syst) MeV/c2, a value that is consistent with theoretical expectations for the previously unseen 1D23 cc¯ meson. We find no other narrow resonance and set upper limits on the branching fractions of the X(3872)→χc1γ and χc2γ decays.