  An interacting quantum system that is subject to disorder may cease to
thermalize due to localization of its constituents, thereby marking the
breakdown of thermodynamics. The key to our understanding of this phenomenon
lies in the system's entanglement, which is experimentally challenging to
measure. We realize such a many-body-localized system in a disordered
Bose-Hubbard chain and characterize its entanglement properties through
particle fluctuations and correlations. We observe that the particles become
localized, suppressing transport and preventing the thermalization of
subsystems. Notably, we measure the development of non-local correlations,
whose evolution is consistent with a logarithmic growth of entanglement entropy
- the hallmark of many-body localization. Our work experimentally establishes
many-body localization as a qualitatively distinct phenomenon from localization
in non-interacting, disordered systems.
