A precision measurement of the mass of the hc(P11) state of charmonium has been made using a sample of 24.5×106 ψ(2S) events produced in e+e- annihilation at the Cornell Electron Storage Ring (CESR). The reaction used was ψ(2S)→π0hc, π0→γγ, hc→γηc, and the reaction products were detected in the CLEO-c detector. Data have been analyzed both for the inclusive reaction and for the exclusive reactions in which ηc decays are reconstructed in 15 hadronic decay channels. Consistent results are obtained in the two analyses. The averaged results of the present measurements are M(hc)=3525.28±0.19(stat.)±0.12(syst.)MeV, and B(ψ(2S)→π0hc)×B(hc→γηc)=(4.19±0. 32±0.45)×10-4. Using the PJ3 centroid mass, ΔMhf(1P) M(χcJ) -M(hc)=+0.02±0.19±0.13MeV.