  We present a precise fully relativistic numerical solution of the two-center
Coulomb problem. The special case of unit nuclear charges is relevant for the
accurate description of the ${\rm H}_2^+$ molecular ion and its isotopologues,
systems that are an active experimental topic. The computation utilizes the
2-spinor minmax approach and the finite-element method. The computed total
energies have estimated fractional uncertainties of a few times $10^{-20}$ for
unit charges and a bond length of 2 atomic units. The fractional uncertainty of
the purely relativistic contribution is $1\times10^{-17}$. The result is
relevant for future precision experiments, whereas at present the uncertainties
arising from the quantum electrodynamic treatment of the rovibrational
transition frequencies. are dominant.
