A universal Gaussian basis set is developed that leads to relativistic Dirac-Fock SCF energies of comparable accuracy as that obtained by the accurate numerical finite-difference method (GRASP2 package) [J. Phys. B 25, 1 (1992)]. The Gaussian-type functions of our universal basis set satisfy the relativistic boundary conditions associated with the finite nuclear model for a finite speed of light and conform to the so-called kinetic balance at the nonrelativistic limit. We attribute the exceptionally high accuracy obtained in our calculations to the fact that the representation of the relativistic dynamics of an electron in a spherical ball finite nucleus near the origin in terms of our universal Gaussian basis set is as accurate as that provided by the numerical finite-difference method. Results of the Dirac-Fock-Coulomb energies for a number of atoms up to No (Z=102) and some negative ions are presented and compared with the recent results obtained with the numerical finite-difference method and geometrical Gaussian basis sets by Parpia, Mohanty, and Clementi [J. Phys. B 25, 1 (1992)]. The accuracy of our calculations is estimated to be within a few parts in 10(9) for all the atomic systems studied.