<p>We consider complete Horn hypergeometric series in two variables and present an algorithm for the determination of their domains of convergence. To this end, we start from the fundamental results due to Horn and we investigate the properties and geometry of the rational algebraic curves delimiting the Reinhardt image of the domain of convergence. Under natural restrictions on the geometry of these curves, we provide an algorithm that iteratively enumerates special subsets of the boundary of the domain of convergence. In particular, we note that the provided algorithm can be efficiently applied to determine the domains of convergence of the analytic continuations of complete hypergeometric series in two variables.</p>