The question of convergence in two-center close-coupling expansions is addressed via the study of positron-hydrogen scattering. It is found that the major cross sections do converge if sufficient number of states from a complete basis, centered separately on the hydrogen atom and positronium, are used to expand the total scattering wave function. The underlying equations are computationally highly ill conditioned, and a simple, numerically efficient technique is given that alleviates the problem. Generally, we find good agreement with available experiment and some previous theory. However, calculations that only used eigenstates for the positronium center yielded cross sections for positronium formation in the 2s and 2p states that are higher than the convergent ones obtained in this work.