  We examine a model of hadronic diffractive scattering which interpolates
between perturbative QCD and non-perturbative fits. We restrict the
perturbative QCD resummation to the large transverse momentum region, and use a
simple Regge-pole parametrization in the infrared region. This picture allows
us to account for existing data, and to estimate the size of the perturbative
contribution to future diffractive measurements. At LHC and SSC energies, we
find that a cut-off BFKL equation can lead to a measurable perturbative
component in traditionally soft processes. In particular, we show that the
total pp cross section could become as large as 228 mb (160 mb) and the rho
parameter as large as 0.23 (0.24) at the SSC (LHC).
