We have measured the ratio of prompt production rates of the charmonium states χc1 and χc2 in 110pb−1 of pp¯ collisions at s√=1.8TeV. The photon from their decay into J/ψγ is reconstructed through conversion into e+e− pairs. The energy resolution this technique provides makes the resolution of the two states possible. We find the ratio of production cross sections σχc2σχc1=0.96±0.27(stat)±0.11(syst) for events with pT(J/ψ)>4.0GeV/c, |η(J/ψ)|<0.6, and pT(γ)>1.0GeV/c.