Ferroelectric thin films and superlattices are currently the subject of intensive research because of the interest they raise for technological applications and also because their properties are of fundamental scientific importance. Ferroelectric superlattices allow the tuning of the ferroelectric properties while maintaining perfect crystal structure and a coherent strain, even throughout relatively thick samples. This tuning is achieved in practice by adjusting both the strain, to enhance the polarization, and the composition, to interpolate between the properties of the combined compounds. Here we show that superlattices with very short periods possess a new form of interface coupling, based on rotational distortions, which gives rise to 'improper' ferroelectricity. These observations suggest an approach, based on interface engineering, to produce artificial materials with unique properties. By considering ferroelectric/paraelectric PbTiO3/SrTiO3 multilayers, we first show from first principles that the ground-state of the system is not purely ferroelectric but also primarily involves antiferrodistortive rotations of the oxygen atoms in a way compatible with improper ferroelectricity. We then demonstrate experimentally that, in contrast to pure PbTiO3 and SrTiO3 compounds, the multilayer system indeed behaves like a prototypical improper ferroelectric and exhibits a very large dielectric constant of epsilon(r) approximately 600, which is also fairly temperature-independent. This behaviour, of practical interest for technological applications, is distinct from that of normal ferroelectrics, for which the dielectric constant is typically large but strongly evolves around the phase transition temperature and also differs from that of previously known improper ferroelectrics that exhibit a temperature-independent but small dielectric constant only.