  We investigate the motion of a tagged spin in a ferromagnetic Ising chain
evolving under Kawasaki dynamics. At equilibrium, the displacement is Gaussian,
with a variance growing as $A t^{1/2}$. The temperature dependence of the
prefactor $A$ is derived exactly. At low temperature, where the static
correlation length $\xi$ is large, the mean square displacement grows as
$(t/\xi^2)^{2/3}$ in the coarsening regime, i.e., as a finite fraction of the
mean square domain length. The case of totally asymmetric dynamics, where $(+)$
(resp. $(-)$) spins move only to the right (resp. to the left), is also
considered. In the steady state, the displacement variance grows as $B
t^{2/3}$. The temperature dependence of the prefactor $B$ is derived exactly,
using the Kardar-Parisi-Zhang theory. At low temperature, the displacement
variance grows as $t/\xi^2$ in the coarsening regime, again proportionally to
the mean square domain length.
