  A calculation of the simplest part of the second-order electron self-energy
(loop after loop irreducible contribution) for hydrogen-like ions with nuclear
charge numbers $3 \leq Z \leq 92$ is presented. This serves as a test for the
more complicated second-order self-energy parts (loop inside loop and crossed
loop contributions) for heavy one-electron ions. Our results are in strong
disagreement with recent calculations of Mallampalli and Sapirstein for low $Z$
values but are compatible with the two known terms of the analytical
$Z\alpha$-expansion.
