We are aiming at predictions for the Lamb shift in hydrogen-like systems with a relative accuracy of10−6.This necessitates calculations of all radiative corrections of first and second order in &agr; but exact in the coupling constantZ&agr;to the external Coulomb field. Results for all known corrections including unclear structure effects are presented for the 1S-ground state in lead82208Pb81+and uranium92238U91+.©1999 American Institute of Physics.