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 increasing as At1/2. The temperature dependence of the prefactor A is derived exactly. At low temperature, where the static correlation length ξ is large, the mean square displacement increases as (t/ξ2)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 increases as Bt2/3. The temperature dependence of the prefactor B is derived exactly, using the Kardar–Parisi–Zhang theory. At low temperature, the displacement variance increases as t/ξ2 in the coarsening regime, again proportionally to the mean square domain length.