Based on our preceding paper, this note is concerned with the exponential stability of the solution semigroup for the abstract linear autonomous functional differential equation x(t) = L(x_t) (*) where L is a continuous linear operator on some abstract phase space B into a Banach space E. We prove that the solution semigroup of (*) is exponentially stable if and only if the fundamental operator (*) is exponentially stable and the phase space B is an exponentially fading memory space.