  We report high-precision calculations of the nuclear recoil effect to the
Lamb shift of hydrogen-like atoms to the first order in the electron-nucleus
mass ratio and to all orders in the nuclear binding strength parameter
$Z\alpha$. The results are in excellent agreement with the known terms of the
$Z\alpha$ expansion and allow an accurate identification of the nonperturbative
higher-order remainder. For hydrogen, the higher-order remainder was found to
be much larger than anticipated. This result resolves the long-standing
disagreement between the numerical all-order and the analytical
$Z\alpha$-expansion approaches to the recoil effect and completely removes the
second-largest theoretical uncertainty in the hydrogen Lamb shift of the $1S$
and $2S$ states.
