We report on the experimental implementation of a spin pump with ultracold bosonic atoms in an optical superlattice. In the limit of isolated double wells it represents a 1D dynamical version of the quantum spin Hall effect. Starting from an antiferromagnetically ordered spin chain, we periodically vary the underlying spin-dependent Hamiltonian and observe a spin current without charge transport. We demonstrate a novel detection method to measure spin currents in optical lattices via superexchange oscillations emerging after a projection onto static double wells. Furthermore, we directly verify spin transport through in-situ measurements of the spins center of mass displacement.