We present the partial-propensity stochastic simulation algorithm with composition-rejection sampling (PSSA-CR). It is an exact formulation of the stochastic simulation algorithm (SSA) for well-stirred systems of coupled chemical reactions. The new formulation is a partial-propensity variant [R. Ramaswamy, N. Gonzalez-Segredo, and I. F. Sbalzarini, J. Chem. Phys. 130, 244104 (2009)] of the composition- rejection SSA [A. Slepoy, A. P. Thompson, and S. J. Plimpton, J. Chem. Phys. 128, 205101 (2008)]. The computational cost of this new formulation is bounded by a constant for weakly coupled reaction networks, and it increases at most linearly with the number of chemical species for strongly coupled reaction networks. PSSA-CR thus combines the advantages of partial-propensity methods and the composition-rejection SSA, providing favorable scaling of the computational cost for all classes of reaction networks.