(1.1) Z= F(z, p, t) It is assumed that for p a=0,c system (1.1) possesses a stable almost periodic solution p(p,,, t). It is further assumed that F(z, 1f, t) for a fixed p is almost periodic in t uniformly with respect to z in a cylindrical neighborhood of p(p,0, t) of radius ;r > 0. Here a function F(z, P, t) for a fixed p is said to be almost periodic in t uniformly with respect to z in a set R if for any s > 0 there exists a relatively dense set S(s) such that if z belongs to R, r belongs to S, then r is an s-translation number of F(p, z, t). Our main result is that for p near po there exists a unique stable almost periodic solution p(t, p) near p(t, po) (Theorem 2.2). In obtaining the above result, we utilize a general geometric approach to our problem based on the properties of translation numbers. In Part III we apply this geometric approach to the generalized pendulum differential equation: