The linearized-augmented-plane-wave (LAPW) method for thin films is generalized by removing the remaining shape approximation to the potential inside the atomic spheres. A new technique for solving Poisson's equation for a general charge density and potential is described and implemented in the film LAPW method. In the resulting full-potential LAPW method (FLAPW), all contributions to the potential are completely taken into account in the Hamiltonian matrix elements. The accuracy of the method---already well known for clean metal surfaces---is demonstrated for the case of a nearly free (noninteracting) ${\mathrm{O}}_{2}$ molecule which is a severe test case of the method because of its large anisotropic charge distribution. Detailed comparisons show that the accuracy of the FLAPW results for ${\mathrm{O}}_{2}$ exceeds that of existing state-of-the-art local-density linear-combination-of-atomic-orbitals (LCAO)-type calculations, and that taking the full potential LAPW results as a reference, the LCAO basis can be improved by adding off-site functions. Thus the full-potential LAPW is a unified method which is ideally suited to test not only molecular adsorption on surfaces, but also the components of the same system separately, i.e., the extreme limits of the molecule and the clean surface.