  We use exact diagonalization to study the breakdown of many-body localization
in a strongly disordered and interacting system coupled to a thermalizing
environment. We show that the many-body level statistics cross over from
Poisson to GOE, and the localized eigenstates thermalize, with the crossover
coupling decreasing with the size of the bath in a manner consistent with the
hypothesis that an infinitesimally small coupling to a thermodynamic bath
should destroy localization of the eigenstates. However, signatures of
incomplete localization survive in spectral functions of local operators even
when the coupling to the environment is non-zero. These include a discrete
spectrum and a gap at zero frequency. Both features are washed out by line
broadening as one increases the coupling to the bath.
