We report measurements of the inclusive reaction, p (p) over bar --> pX (p) over bar, in events where either or both the beam-like final-state baryons were detected in Roman-pot spectrometers and the central system was detected in the UA2 calorimeter. A Double-Pomeron-Exchange (DPE) analysis of these data and single diffractive data from the same experiment demonstrates that, for central masses of a few GeV, the extracted Pomeron-Pomeron total cross section, sigma(pp)(tot), exhibits an enhancement which exceeds factorization expectations by an order-of-magnitude. This may be a signature for glueball production. The enhancement is shown to be independent of uncertainties connected with possible non-universality of the Pomeron flux factor. Based on our analysis, we present DPE cross section predictions, for unit (1 mb) Pomeron-Pomeron total cross section, at the Tevatron, LHC and the 920 GeV fixed-target experiment, HERA-B.