Gánti's chemoton model is an illustrious example of a minimal cell model. It is composed of three stoichiometrically coupled autocatalytic subsystems: a metabolism, a template replication process, and a membrane enclosing the other two. Earlier studies on chemoton dynamics yield inconsistent results. Furthermore, they all appealed to deterministic simulations, which do not take into account the stochastic effects induced by small population sizes. We present, for the first time, results of a chemoton simulation in which these stochastic effects have been taken into account. We investigate the dynamics of the system and analyze in depth the mechanisms responsible for the observed behavior. Our results suggest that, in contrast to the most recent study by Munteanu and Solé, the stochastic chemoton reaches a unique stable division time after a short transient phase. We confirm the existence of an optimal template length and show that this is a consequence of the monomer concentration, which depends on the template length and the initiation threshold. Since longer templates imply shorter division times, these results motivate the selective pressure toward longer templates observed in nature.