Authors’ abstract: We consider a mild solution $v_f(\cdot, 0)$ of a well-posed inhomogeneous Cauchy problem $\dot v(t)= A(t) v(t)+ f(t)$, $v(0)= 0$, on a complex Banach space $X$, where $A(\cdot)$ is a 1-periodic operator-valued function. We prove that if $v_f(\cdot,0)$ belongs to $AP_0(\bbfR_+,X)$, for each $f\in AP_0(\bbfR_+, X)$ then for each $x\in X$ the solution of the well-posed Cauchy problem $\dot u(t)= A(t) v(t)$, $u(0)= x$, is uniformly exponentially stable. The converse statement is also true. [$AP_0(\bbfR_+,X)$ denotes the smallest closed subspace (of the Banach space of $X$-valued, bounded and uniformly continuous functions on $\bbfR_+$) which contains the almost periodic functions as well as those functins which are zero on some interval $[0,t]$ and identical with an almost periodic function on $[t,\infty)$]. Our approach is based on the spectral theory of evolution semigroups.