  We present a theoretical method to calculate Delbr\"uck scattering amplitudes
for photon energies above the electron-positron pair production threshold. The
method is based on the application of the relativistic Dirac-Coulomb Green
function and describes the interaction of the virtual $e^+e^-$ pair with the
Coulomb field of a target to all orders in the coupling strength parameter
$\alpha Z$. To illustrate the application of the developed approach, detailed
calculations have been performed for the scattering of 2.754~MeV photons off
bare ions with a wide range of nuclear charge numbers. Results of these
calculations clearly indicate that the higher-order terms beyond the Born
approximation lead to a strong enhancement of the imaginary part of the
Delbr\"uck amplitude and have to be taken into account for the analysis and
guidance of gamma-ray scattering experiments.
