Organic light-emitting diodes (OLEDs) promise highly efficient lighting and display technologies. We introduce a new class of linear donor-bridge-acceptor light-emitting molecules, which enable solution-processed OLEDs with near-100% internal quantum efficiency at high brightness. Key to this performance is their rapid and efficient utilization of triplet states. Using time-resolved spectroscopy, we establish that luminescence via triplets occurs within 350 nanoseconds at ambient temperature, after reverse intersystem crossing to singlets. We find that molecular geometries exist at which the singlet-triplet energy gap (exchange energy) is close to zero, so that rapid interconversion is possible. Calculations indicate that exchange energy is tuned by relative rotation of the donor and acceptor moieties about the bridge. Unlike other systems with low exchange energy, substantial oscillator strength is sustained at the singlet-triplet degeneracy point.