Gaussian wave functions with optimized nonlinear variational parameters are applied to evaluate clamped-nucleus nonrelativistic energies as well as the lowest order relativistic corrections for the ground states of the hydrogen atom, hydrogen molecular ion H-2(+), helium atom, and hydrogen molecule. The two-electron functions used depend explicitly on the interelectronic distance r(12) but do not describe the cusp properly. Despite this, for H-2(+) and H-2 the results are more accurate than ever reported, and for He they are inferior only to the best calculations employing Hylleraas-type expansions. It is demonstrated that, contrary to a common opinion, Gaussian wave functions are very well suited for high-accuracy relativistic computations even in the Breit-Pauli approximation, provided that the nonlinear parameters are optimized with respect to the nonrelativistic energy.