  We calculate the two-loop Bethe logarithm correction to atomic energy levels
in hydrogen-like systems. The two-loop Bethe logarithm is a low-energy quantum
electrodynamic (QED) effect involving multiple summations over virtual excited
atomic states. Although much smaller in absolute magnitude than the well-known
one-loop Bethe logarithm, the two-loop analog is quite significant when
compared to the current experimental accuracy of the 1S-2S transition: it
contributes -8.19 and -0.84 kHz for the 1S and the 2S state, respectively. The
two-loop Bethe logarithm has been the largest unknown correction to the
hydrogen Lamb shift to date. Together with the ongoing measurement of the
proton charge radius at the Paul Scherrer Institute its calculation will bring
theoretical and experimental accuracy for the Lamb shift in atomic hydrogen to
the level of 10^(-7).
