We report the development of the fully relativistic convergent close-coupling method based on the solution of the Dirac equation. A complete square-integrable Dirac L-spinor basis is used to obtain a set of target states spanning the target discrete and continuous positive- and negative-energy spectra. The present implementation is for quasi-one-electron atoms whose electronic configuration corresponds to the first column of the periodic table. By way of example, we consider elastic scattering of 7 eV electrons on the ground state of cesium, where the full set of spin asymmetries (A1, A2, Ann) has been measured. Excellent agreement with experiment is found.