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The realization of high-transition-temperature (high-T_c) superconductivity confined to nanometre-sized interfaces has been a long-standing goal because of potential applications^, and the opportunity to study quantum phenomena in reduced dimensions^,. This has been, however, a challenging target: in conventional metals, the high electron density restricts interface effects (such as carrier depletion or accumulation) to a region much narrower than the coherence length, which is the scale necessary for superconductivity to occur. By contrast, in copper oxides the carrier density is low whereas T_c is high and the coherence length very short, which provides an opportunity—but at a price: the interface must be atomically perfect. Here we report superconductivity in bilayers consisting of an insulator (La₂CuO₄) and a metal (La_1.55Sr_0.45CuO₄), neither of which is superconducting in isolation. In these bilayers, T_c is either …