  We demonstrate that the finiteness of the limiting values of the lower energy
levels of a hydrogen atom under an unrestricted growth of the magnetic field,
into which this atom is embedded, is achieved already when the vacuum
polarization (VP) is calculated in the magnetic field within the approximation
of the local action of Euler--Heisenberg. We find that the mechanism for this
saturation is different from the one acting, when VP is calculated via the
Feynman diagram in the Furry picture. We study the effective potential that
appears when the adiabatic (diagonal) approximation is exploited for solving
the Schr\"{o}dinger equation for the longitudinal degree of freedom of the
electron on the lowest Landau level in the atom. We find that the (effective)
potential of a point-like charge remains nonsingular thanks to the growing
screening provided by VP. The regularizing length turns out to be $\sqrt{\alpha
/3\pi }\lambdabar_{\mathrm{C}}$, where $\lambdabar_{\mathrm{C}}$ is the
electron Compton length. The family of effective potentials, labeled by growing
values of the magnetic field condenses towards a certain limiting,
magnetic-field-independent potential-distance curve. The~limiting values of
even ground-state energies are determined for four magnetic quantum numbers
using the Karnakov--Popov method.
