Ordered atomic-scale superlattices on surface hold great interest both for basic science and for potential applications in advanced technology. However, controlled fabrication of superlattices down to atomic scale has proven exceptionally challenging. Here we demonstrate the segregation-growth and self-organization of ordered S atomic superlattices confined at the interface between graphene and S-rich Cu substrates. Scanning tunneling microscope (STM) studies show that, by finely controlling the growth temperature, we obtain well-ordered S (sub)nanometer-cluster superlattice and monoatomic superlattices with various periods at the interface. These atomic superlattices are stable in atmospheric environment and robust even after high-temperature annealing (~ 350 oC). Our experiments demonstrate that the S monoatomic superlattice can drive graphene into the electronic Kekulé distortion phase when the period of the ordered S adatoms is commensurate with graphene lattice. Our results not only open a road to realize atomic-scale superlattices at interfaces, but also provide a new route to realize exotic electronic states in graphene.