We have applied a discretized version of the generator coordinate Hartree-Fock method to generate a universal Gaussian basis set for the atoms from H through Lr. In general, for He to Rn the ground state Hartree-Fock total energy results obtained with our universal Gaussian basis set are lower than the corresponding ones obtained by Clementi et al. using larger atom-optimized geometrical Gaussian basis sets. For the atoms from H through Lr we have obtained ground state energy values with at least seven digits of accuracy (except for Pm, with six digits), in comparison with the corresponding ten digits of the numerical Hartree-Fock total energy results.