  We examine what happens when a strongly many body localized system is coupled
to a weak heat bath, with both system and bath containing similar numbers of
degrees of freedom. Previous investigations of localized systems coupled to
baths operated in regimes where the back action of the system on the bath is
negligible, and concluded that the bath generically thermalizes the system. In
this work we show that when the system is strongly localized and the bath is
only weakly ergodic, the system can instead localize the bath. We demonstrate
this both in the limit of weak coupling between system and bath, and in the
limit of strong coupling, and for two different types of `weak' bath - baths
which are close to an atomic limit, and baths which are close to a
non-interacting limit. The existence of this `many body localization proximity
effect' indicates that many body localization is more robust than previously
appreciated, and can not only survive coupling to a (weak) heat bath, but can
even destroy the bath.
