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We have measured the ratio of prompt production rates of the charmonium states χ c1 and χ c2 in 110 pb-1 of pp̅ collisions at √s= 1.8 TeV. 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 (σ χ c2/ σ χ c1=0.96 ± 0.27(stat) ± 0.11(syst) for events with pT(J/ψ)> 4.0 GeV/c, |η(J/ψ)|, 0.6, and pT( γ )>g> 1.0 GeV/c.