The coolest known brown dwarfs are our best analogs to extrasolar gas-giant planets. The prolific detections of such cold substellar objects in the past 2 years have spurred intensive follow-up, but the lack of accurate distances is a key gap in our understanding. We present a large sample of precise distances based on homogeneous mid-infrared astrometry that robustly establishes absolute fluxes, luminosities, and temperatures. The coolest brown dwarfs have temperatures of 400 to 450 kelvin and masses almost equal to 5 to 20 times that of Jupiter, showing they bridge the gap between hotter brown dwarfs and gas-giant planets. At these extremes, spectral energy distributions no longer follow a simple correspondence with temperature, suggesting an increasing role of other physical parameters, such as surface gravity, vertical mixing, clouds, and metallicity.