Thermalization is a ubiquitous process of statistical physics, in which a physical system reaches an equilibrium state that is defined by a few global properties such as temperature. Even in isolated quantum many-body systems, limited to reversible dynamics, thermalization typically prevails(1). However, in these systems, there is another possibility: many-body localization (MBL) can result in preservation of a non-thermal state(2,3). 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 spatially increasing field-but no disorder-can also exhibit MBL4, resulting in Stark MBL (5). 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. Furthermore, by engineering a varying gradient, we create a disorder-free system with coexisting long-lived thermalized and non-thermal 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. Experiments with a trapped-ion quantum simulator observe Stark many-body localization, in which the quantum system evades thermalization despite having no disorder.