An integral equation theory is presented for the pair correlation functions and phase behavior of symmetric nonadditive hard sphere mixtures with hard sphere diameters given by sigma(AA) = sigma(BB) = lambda d and sigma(AB) = d. This mixture exhibits a fluid-fluid phase separation into an A-rich phase and a B-rich phase at high densities. The theory incorporates, into the closure approximation, all terms that can be calculated exactly in the density expansion of the direct correlation functions. We find that the closure approximation developed in this work is accurate for the structure and phase behavior over the entire range of lambda, when compared to computer simulations, and is significantly more accurate than the previous theories.