  We present a theoretical method to calculate Delbr\"uck scattering
amplitudes. Our formalism is based on the exact analytical Dirac-Coulomb
Green's function and, therefore, accounts for the interaction of the virtual
electron-positron pair with the nucleus to all orders, including the Coulomb
corrections. The numerical convergence of our calculations is accelerated by
solving the radial integrals that are involved analytically in the asymptotic
region. Numerical results for the collision of photons with energies 102.2 keV
and 255.5 keV with bare neon and lead nuclei are compared with the predictions
of the lowest-order Born approximation. We find that our method can produce
accurate results within a reasonable computation time and that the Coulomb
corrections enhance the absolute value of the Delbr\"uck amplitude by a few
percent for the studied photon energies.
