The probability of internal gamma-ray conversion is theoretically investigated for hydrogen-like ions versus the corresponding neutral atoms. The relevant calculations are performed by the relativistic Dirac-Fock method. The results reveal that the effect of multiple ionization on the coefficients of internal conversion in the K shell is maximal near the ionization threshold and for transitions of high multipole order, where this effect can be as great as a few orders of magnitude. The distinction between the coefficient of internal conversion in the K shell of a neutral atom and that in the respective hydrogen-like ion decreases with increasing transition energy, but it remains sizable for transitions of practical importance. It is found that the ionization of an atom to a hydrogen-like ion with allowance for conversion in external atomic shells may change significantly (by up to eight orders of magnitude) the lifetime of the nucleus being considered. The predicted effects can be observed in experiments with beams of relativistic heavy ions.