  We calculate nuclear-polarization energy shifts for hydrogenlike
$^{208}_{~82}$Pb$^{81+}$. A retarded transverse part as well as the Coulomb
part is taken into account as the electromagnetic interaction between an
electron and the nucleus. With a finite charge distribution for the nuclear
ground state and the random-phase approximation to describe the nuclear
excitations, we obtain nuclear polarization energy of the $1s_{1/2}$ state as
--38.2 (--37.0) meV in the Feynman (Coulomb) gauge. For the $2s_{1/2}$,
$2p_{1/2}$ and $2p_{3/2}$ states, they are --6.7 (--6.4), --0.2 (--0.2) and
+0.0 (+0.0) meV, respectively. The seagull term in the two-photon exchange
diagrams is shown to be quite important to obtain the gauge invariance of the
nuclear polarization energies.
