  The $g$ factor of the $4_1^+$ state in $^{68}$Zn has been remeasured with
improved energy resolution of the detectors used. The value obtained is
consistent with the previous result of a negative $g$ factor thus confirming
the dominant $0g_{9/2}$ neutron nature of the $4_1^+$ state. In addition, the
accuracy of the $g$ factors of the $2_1^+$, $2_2^+$ and $3_1^-$ states has been
improved an d their lifetimes were well reproduced. New large-scale shell model
calculations based on a $^{56}$Ni core and an $0f_{5/2}1pg_{9/2}$ model space
yield a theoretical value, $g(4_1^+) = +0.008$. Although the calculated value
is small, it cannot fully explain the experimental value, $g(4_1^+) =
-0.37(17)$. The magnitude of the deduced B(E2) of the $4_1^+$ and $2_1^+$
transition is, however, rather well described. These results demonstrate again
the importance of $g$ factor measurements for nuclear structure determination s
due to their specific sensitivity to detailed proton and neutron components in
the nuclear wave functions.
