We have observed exclusive gammagamma production in proton-antiproton collisions at sqrt{s}=1.96 TeV, using data from 1.11 pm 0.07 fb^{-1} integrated luminosity taken by the Run II Collider Detector at Fermilab. We selected events with two electromagnetic showers, each with transverse energy E_T > 2.5 GeV and pseudorapidity |eta| < 1.0, with no other particles detected in -7.4 < eta < +7.4. The two showers have similar E_T and azimuthal angle separation Deltaphi sim pi; 34 events have two charged particle tracks, consistent with the QED process p bar{p} to p + e^+e^- + bar{p} by two-photon exchange, while 43 events have no charged tracks. The number of these events that are exclusive pi^0pi^0 is consistent with zero and is < 15 at 95% C.L. The cross section for pbar{p} to p+gammagamma+bar{p} with |eta(gamma)| < 1.0 and E_T(gamma) > 2.5$ GeV is 2.48^{+0.40}_{-0.35}(stat)^{+0.40}_{-0.51}(syst) pb.
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