We experimentally realize a method to produce nonequilibrium Bose-Einstein condensates with condensed fraction exceeding those of equilibrium samples with the same parameters. To do this, we immerse an ultracold Bose gas of ^{87}Rb in a cloud of ^{39}K with substantially higher temperatures, providing a controlled source of dissipation. By combining the action of the dissipative environment with evaporative cooling, we are able to progressively distil the nonequilibrium Bose-Einstein condensate from the thermal cloud. We show that by increasing the strength of the dissipation it is even possible to produce condensates above the critical temperature. We finally demonstrate that our out-of-equilibrium samples are long lived and do not reach equilibrium in a time that is accessible for our experiment. Due to its high degree of control, our distillation process is a promising tool for the engineering of open quantum systems.