We report on the recent evaluation of the two-photon electron self energy t
o all orders in the interaction with the Coulomb field of the nucleus. With
the present results at hand the major theoretical uncertainty is diminishe
d, which provides predictions of the ground-state energy with a relative ac
curacy of about 10(-6) for the hydrogenlike uranium and lead systems. This
allows for high-precision tests of quantum electrodynamics (QED) in strong
fields that are expected to be experimentally available in the near future.