A search for new physics has been carried out in the channel p¯p→γγ+ET, as expected in various supersymmetric models. We require two photons with transverse energy EγT>12GeV and pseudorapidity |ηγ|<1.1. The distribution of missing transverse energy ( ET) is consistent with background and no events have ET>25GeV. We set limits on production cross sections for selectron, sneutrino, and neutralino pairs, decaying into photons, which range from about 400 fb to 1 pb depending on the sparticle masses. A general limit of 185 fb (95% C.L.) is set on σ(p¯p→γγET+X), where EγT>12GeV, |ηγ|<1.1, and ET>25GeV.Received 12 December 1996DOI:https://doi.org/10.1103/PhysRevLett.78.2070©1997 American Physical Society