Semiconductor quantum dots are considered to be the leading venue for fabricating on-demand sources of single photons. However, the generation of long-lived dark excitons imposes significant limits on the efficiency of these sources. We demonstrate a technique that optically pumps the dark exciton population and converts it to a bright exciton population, using intermediate excited biexciton states. We show experimentally that our method considerably reduces the DE population while doubling the triggered bright exciton emission, approaching thereby near-unit fidelity of quantum dot depletion.