This dissertation presents the results of a search for supersymmetry in protonantiproton collisions with a center of mass energy of 1.96 TeV studied with the Collider Detector at Fermilab. Our strategy is to select collisions with two photons in the nal state that have the properties of being the decays of very massive supersymmetric particles. This includes looking for large total energy from the decayed particles as well as for the presence of particles that leave the detector without interacting. We nd no events using 2.6 fb-1 of data collected during the 2004-2008 collider run of the Fermilab Tevatron which is consistent with the background estimate of 1.4 +/- 0.4 events. Since there is no evidence of new particles we set cross section limits in a gaugemediated supersymmetry model with e 01 ! eG, where the e 01 and eG are the lightest neutralino and the gravitino (the lightest supersymmetric particle), respectively. We set limits on models as a function of the e 01 mass and lifetime, producing the world's most sensitive search for e 01by excluding masses up to 149 GeV=c2 for e 01 lifetimes much less than 1 ns.