Coupling-constant integration of the mutual Coulomb interaction of a many-electron system formally gives the density functional exchange-correlation energy. The exchange-energy scales linearly with the coupling-constant, but the dependence of the correlation energy on the coupling-constant is not known. We present accurate numerical results for the coupling-constant dependence of the density functional correlation energy for two-electron systems. For the systems studied, we find that the coupling-constant dependence of the correlation energy can be represented accurately by a cubic polynomial or a two parameter Pade approximation for coupling constants between zero and one. We also find that the recently published approximate correlation energy of Perdew, Burke and Ernzerhof [Phys. Rev. Lett. 77, 3865 (1996)] follows the exact dependence remarkably well.