  Under the assumption of total absorption and dominance of the imaginary part
of the scattering amplitude, we present a sum rule for any hadronic elastic
differential cross-section ${d\sigma}\over{dt}$: the dimensionless quantity
${{1}\over{2}}\int (dt)\sqrt{{{1}\over {\pi}}{{d\sigma}\over{dt}}} \to 1$, at
asymptotic energies. Experimental data from ISR and Tevatron confirm a trend
towards its saturation and some estimates are presented for LHC. Its
universality and further consequences for the nature of absorption in QCD based
models for elastic and total cross-sections are explored.
