A general analysis of the two-body Dirac equation is presented for the case of equal masses interacting via a static Coulomb potential. Radial equations are derived and their analytical structure is discussed. Standard analytical and perturbative methods have failed to provide solutions to the radial equations due to the presence of the singularity on the negative radial axis at roughly the distance of the classical electron radius. The exact radial equations are solved using finite-element analysis, and the low-lying bound states are obtained to an accuracy of one part in 1018. The effect of the singularity is clearly seen in the structure of the finite-element radial components.