Consider the Duffing differential equation \[ d^2x/dt^2- x+ x^3= f(t),\tag{\(*\)} \] where \(f\) is almost periodic. By using the theory of exponential dichotomy, the author first proves that \((*)\) has a unique bounded solution provided \(|f|\leq 8/27\). Then, by constructing a Lyapunov function, the existence of a unique almost periodic solution \(x_f(t)\) of \((*)\) is established satisfying \(|x_f|\leq{1\over 3}\) if \(|f|\leq{8\over 27}\).