In the path-following problem formulated in this note, it is required that the error between the system output and the desired geometric path eventually be less than any prespecified constant. If in a nonlinear multiple-input-multiple-output (MIMO) system the output derivatives do not enter into its zero dynamics, a condition relating path geometry and stabilizability of the zero dynamics is given under which a solution to this problem exists. The solution is obtained by combining input-to-state stability and hybrid system methodologies.