The differential cross section for Delbrück scattering has been measured at photon energies between 1 and 7 GeV and scattering angles between 1 and 3 mrad on copper, silver, gold, and uranium targets. The results confirm the predictions of quantum electrodynamics, if the exchange of a very large number of photons with the nucleus (Coulomb correction) is taken into account. At momentum transfers of a few MeV/c, the Coulomb correction for uranium results in a reduction of the cross section by a factor between 3 and 5 as compared to the prediction of lowest-order relativistic perturbation theory. The photon-splitting process has been experimentally detected at the same energies and angles. Estimates of the cross section are given.