We have searched for exclusive {gamma}{gamma} production in proton-antiproton collisions at {radical}(s)=1.96 TeV, using 532 pb{sup -1} of integrated luminosity taken by the run II Collider Detector at Fermilab. The event signature requires two electromagnetic showers, each with transverse energy E{sub T}>5 GeV and pseudorapidity |{eta}|<1.0, with no other particles detected in the event. Three candidate events are observed. We discuss the consistency of the three events with {gamma}{gamma}, {pi}{sup 0}{pi}{sup 0}, or {eta}{eta} production. The probability that other processes fluctuate to {>=}3 events is 1.7x10{sup -4}. An upper limit on the cross section of pp{yields}p+{gamma}{gamma}+p production is set at 410 fb with 95% confidence level.