  Accurate calculations of the nuclear recoil effect to the Lamb shift of
hydrogen-like atoms are presented. Numerical results are reported for the $ns$
states with $n \leq 5$ and for the $2p_{1/2}$ and $2p_{3/2}$ states. The
calculations are performed to the first order in the electron-nucleus mass
ratio and to all orders in the nuclear binding strength parameter $Z\alpha$
(where $Z$ is the nuclear charge number and $\alpha$ is the fine structure
constant). The obtained results provide accurate predictions for the
higher-order remainder beyond the known $Z\alpha$-expansion terms. In the case
of hydrogen, the remainder was found to be much larger than anticipated. This
result resolves the previously reported disagreement between the numerical
all-order and the analytical $Z\alpha$-expansion approaches for the nuclear
recoil effect in the hydrogen Lamb shift.
