We present the first observation of exclusive e(+)e(-) production in hadron-hadron collisions, using pp[over] collision data at (square root) s = 1.96 TeV taken by the run II Collider Detector at Fermilab, and corresponding to an integrated luminosity of 532 pb(-1). We require the absence of any particle signatures in the detector except for an electron and a positron candidate, each with transverse energy E(T) > 5 GeV and pseudorapidity |eta| < 2. With these criteria, 16 events are observed compared to a background expectation of 1.9+/-0.3 events. These events are consistent in cross section and properties with the QED process pp[over] --> p + e(+)e(-) + p[over] through two-photon exchange. The measured cross section is 1.6(-0.3)(+0.5)(stat) +/- 0.3(syst) pb. This agrees with the theoretical prediction of 1.71+/-0.01 pb.