  Thermalization is a ubiquitous process of statistical physics, in which
details of few-body observables are washed out in favor of a featureless steady
state. Even in isolated quantum many-body systems, limited to reversible
dynamics, thermalization typically prevails. However, in these systems, there
is another possibility: many-body localization (MBL) can result in preservation
of a non-thermal state. While disorder has long been considered an essential
ingredient for this phenomenon, recent theoretical work has suggested that a
quantum many-body system with a uniformly increasing field -- but no disorder
-- can also exhibit MBL, resulting in `Stark MBL.' Here we realize Stark MBL in
a trapped-ion quantum simulator and demonstrate its key properties: halting of
thermalization and slow propagation of correlations. Tailoring the interactions
between ionic spins in an effective field gradient, we directly observe their
microscopic equilibration for a variety of initial states, and we apply
single-site control to measure correlations between separate regions of the
spin chain. Further, by engineering a varying gradient, we create a
disorder-free system with coexisting long-lived thermalized and nonthermal
regions. The results demonstrate the unexpected generality of MBL, with
implications about the fundamental requirements for thermalization and with
potential uses in engineering long-lived non-equilibrium quantum matter.
