Supernova (SN) explosions are crucial engines driving the evolution of galaxies by shock heating gas, increasing the metallicity, creating dust, and accelerating energetic particles. In 2012 we used the Atacama Large Millimeter/Submillimeter Array to observe SN 1987A, one of the best-observed supernovae since the invention of the telescope. We present spatially resolved images at 450 μm, 870 μm, 1.4 mm, and 2.8 mm, an important transition wavelength range. Longer wavelength emission is dominated by synchrotron radiation from shock-accelerated particles, shorter wavelengths by emission from the largest mass of dust measured in a supernova remnant (>0.2 M Ȯ). For the first time we show unambiguously that this dust has formed in the inner ejecta (the cold remnants of the exploded star's core). The dust emission is concentrated at the center of the remnant, so the dust has not yet been affected by the shocks. If a significant fraction survives, and if SN 1987A is typical, supernovae are important cosmological dust producers. © 2014. The American Astronomical Society. All rights reserved.