This paper analyzes the age of information in a single-source status update system with energy constraints. Specifically, a source provides status updates to a destination through a server assumed to have a battery with finite capacity that is replenished by harvesting energy. Arrival times of the status updates at the source and energy units at the server are assumed to be random according to independent Poisson processes, and service times are assumed to be exponentially distributed and independent of the status and energy arrivals. Average age expressions are derived under the assumption that new status updates from the source preempt a prior status update in service. Numerical results are provided to illustrate the operating regimes where a server that allows preemption of status update packets in service achieves a lower average age compared to a server that does not allow preemption.