We present a convergent variational basis-set calculational scheme for elastic scattering of the positronium atom by the hydrogen atom in S wave. Highly correlated trial functions with appropriate symmetry are needed to achieve convergence. We report convergent results for scattering lengths in atomic units for both singlet ( = 3.49 ± 0.20) and triplet ( = 2.46 ± 0.10) states.