Topological insulators are an emerging class of materials that host highly robust in-gap surface or interface states while maintaining an insulating bulk(1,2). Most advances in this field have focused on topological insulators and related topological crystalline insulators(3) in two dimensions(4-6) and three dimensions(7-10), but more recent theoretical work has predicted the existence of one-dimensional symmetry-protected topological phases in graphene nanoribbons (GNRs)(11). The topological phase of these laterally confined, semiconducting strips of graphene is determined by their width, edge shape and terminating crystallographic unit cell and is characterized by a Z(2) invariant(12) (that is, an index of either 0 or 1, indicating two topological classes-similar to quasi-one-dimensional solitonic systems(13-16)). Interfaces between topologically distinct GNRs characterized by different values of Z(2) are predicted to support half-filled, in-gap localized electronic states that could, in principle, be used as a tool for material engineering(11). Here we present the rational design and experimental realization of a topologically engineered GNR superlattice that hosts a one-dimensional array of such states, thus generating otherwise inaccessible electronic structures. This strategy also enables new end states to be engineered directly into the termini of the one-dimensional GNR superlattice. Atomically precise topological GNR superlattices were synthesized from molecular precursors on a gold surface, Au(111), under ultrahigh-vacuum conditions and characterized by low-temperature scanning tunnelling microscopy and spectroscopy. Our experimental results and first-principles calculations reveal that the frontier band structure (the bands bracketing filled and empty states) of these GNR superlattices is defined purely by the coupling between adjacent topological interface states. This manifestation of non-trivial one-dimensional topological phases presents a route to band engineering in one-dimensional materials based on precise control of their electronic topology, and is a promising platform for studies of one-dimensional quantum spin physics.