In atomic physics, the coherent coupling of a broad and a narrow resonance leads to quantum interference and provides the general recipe for electromagnetically induced transparency (EIT). A sharp resonance of nearly perfect transmission can arise within a broad absorption profile. These features show remarkable potential for slow light, novel sensors and low-loss metamaterials. In nanophotonics, plasmonic structures enable large field strengths within small mode volumes. Therefore, combining EIT with nanoplasmonics would pave the way towards ultracompact sensors with extremely high sensitivity. Here, we experimentally demonstrate a nanoplasmonic analogue of EIT using a stacked optical metamaterial. A dipole antenna with a large radiatively broadened linewidth is coupled to an underlying quadrupole antenna, of which the narrow linewidth is solely limited by the fundamental non-radiative Drude damping. In accordance with EIT theory, we achieve a very narrow transparency window with high modulation depth owing to nearly complete suppression of radiative losses.