  We experimentally realize a method to produce non-equilibrium 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 87Rb in a
cloud of 39K 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
non-equilibrium 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.
