Starting with a bare nucleon-nucleon interaction, for the first time the full relativistic Brueckner—Hartree—Fock equations are solved for finite nuclei in a Dirac—Woods—Saxon basis. No free parameters are introduced to calculate the ground-state properties of finite nuclei. The nucleus 16O is investigated as an example. The resulting ground-state properties, such as binding energy and charge radius, are considerably improved as compared with the non-relativistic Brueckner—Hartree—Fock results and much closer to the experimental data. This opens the door for ab initio covariant investigations of heavy nuclei.