To obtain the maximum luminous efficiency from an organic material, it is necessary to harness both the spin-symmetric and anti-symmetric molecular excitations (bound electron-hole pairs, or excitons) that result from electrical pumping. This is possible if the material is phosphorescent, and high efficiencies have been observed in phosphorescent organic light-emitting devices. However, phosphorescence in organic molecules is rare at room temperature. The alternative radiative process of fluorescence is more common, but it is approximately 75% less efficient, due to the requirement of spin-symmetry conservation. Here, we demonstrate that this deficiency can be overcome by using a phosphorescent sensitizer to excite a fluorescent dye. The mechanism for energetic coupling between phosphorescent and fluorescent molecular species is a long-range, non-radiative energy transfer: the internal efficiency of fluorescence can be as high as 100%. As an example, we use this approach to nearly quadruple the efficiency of a fluorescent red organic light-emitting device.