We have searched for the two-photon production of the c, 0, and 2 charmonium states at the SLAC e+e- collider PEP in the channels K± KS0 , K+K-+-, +- + -, and K+K-K+K-. We identify four c candidates in the K+K-K+K- channel on a negligible background; the one event among them implies a 95%-confidence-level (C.L.) lower limit for "( c) of 1.7 keV. In the other channels we find no evidence for any of the three states. We establish 95%-C.L. upper limits "(c)<15.5 keV, "(0)<17.0 keV, and " (2)<4.2 keV. From all channels combined, we obtain the value "(c)=6.4-3.4+5.0 keV.