Highly correlated positioning of ions underlies Coulomb interactions between ions and electrified interfaces within dense ionic fluids such as biological cells and ionic liquids. Recent work has shown that highly correlated ionic systems behave differently than dilute electrolyte solutions, and interest is focused upon characterizing the electrical and structural properties of the dense electrical double layers (EDLs) formed at internal interfaces. It has been a challenge for experiments to characterize the progressive development of the EDL on the nanoscale as the interfacial electric potential is varied over a range of positive and negative values. Here we address this challenge by measuring X-ray reflectivity from the interface between an ionic liquid (IL) and a dilute aqueous electrolyte solution over a range of interfacial potentials from -450 to 350 mV. The growth of alternately charged cation-rich and anion-rich layers was observed along with a polarity reversal of the layers as the potential changed sign. These data show that the structural development of an ionic multilayer-like EDL with increasing potential is similar to that suggested by phenomenological theories and MD simulations, although our data also reveal that the excess charge beyond the first ionic layer decays more rapidly than predicted.