  The nuclear shape correction to the g factor of a bound electron in 1S-state
is calculated for a number of nuclei in the range of charge numbers from Z=6 up
to Z=92. The leading relativistic deformation correction has been derived
analytically and also its influence on one-loop quantum electrodynamic terms
has been evaluated. We show the leading corrections to become significant for
mid-Z ions and for very heavy elements to even reach the 10^(-6) level.
