  We report calculations of Delbr\"uck scattering that include all-order
Coulomb corrections for photon energies above the threshold of
electron-positron pair creation. Our approach is based on the application of
the Dirac-Coulomb Green function and accounts for the interaction between the
virtual electron-positron pair and the nucleus to all orders in the nuclear
binding strength parameter $\alpha Z$. Practical calculations are performed for
the scattering of 2.754 MeV photons off plutonium atoms. We find that including
the Coulomb corrections enhances the scattering cross section by up to 50% in
this case. The obtained results resolve the long-standing discrepancy between
experimental data and theoretical predictions and demonstrate that an accurate
treatment of the Coulomb corrections is crucial for the interpretation of
existing and guidance of future Delbr\"uck scattering experiments on heavy
atoms.
