A scalar inflaton coupled to a dark photon can produce a large density of the latter. I will discuss the mechanism underlying this production, which takes place mostly at the end of inflation. After reheating, the dark photons are relativistic at first and then redshift to non relativistic. Their energy density today can match that of the observed dark matter in a wide region of parameters of the model. It is interesting to study in detail also the dark photon power spectrum and its cosmic evolution, which predicts a clumpy nature of this dark matter candidate, possibly useful for experimental searches.