Using the CLEO detector at the Cornell Electron Storage Ring, we have searched for two-photon production of charmonium in four-track final states. We have measured Γγγ(ηc)=5.9+2.1-1.8±1.9 keV, and we have obtained 95% CL upper limits of Γγγ(Xc0)<6.2 keV and Γγγ(Xc2)<1.0 keV. © 1990.