The crossover and the ground-state properties in the attractive Hubbard model are studied using the exact Bethe-ansatz equations and the self-consistent field (SCF) approach with renormalized chemical potential. The exact and the SCF results for the ground-state energy, the concentration of double occupied sites D, the chemical potential μ and the kinetic energy are in good agreement for all band fillings n and a wide range of the coupling strength -U/t. The SCF theory in one dimension suggests a smooth crossover from the BCS superconductivity to the superfluidity of local pairs under variation of -U/t and n. We found a simple relationship between D, n and the BCS order parameter.