The success of metal halide perovskites in photovoltaic and light-emitting diodes (LEDs) motivates their application as a solid-state thin-film laser. Various perovskites have shown optically pumped stimulated emission of lasing and amplified spontaneous emission (ASE), yet the ultimate goal of electrically pumped stimulated emission has not been achieved. As an essential step toward this goal, here, a perovskite diode structure that simultaneously exhibits stable operation at high current density (≈1 kA cm-2 ) and optically excited ASE (with a threshold of 180 µJ cm-2 ) is reported. This diode structure achieves an electroluminescence quantum efficiency of 0.8% at 850 A cm-2 , which is estimated to be ≈3% of the charge carrier population required to reach ASE in the same device. It is shown that the formation of a large angle waveguide mode and the reduction of parasitic absorption losses are two major design principles for diodes to obtain a positive gain for stimulated emission. In addition to its prospect as a perovskite laser, a new application of electrically pumped ASE is proposed as an ideal perovskite LED architecture allowing 100% external radiation efficiency.