The present paper is a continuation of ref. 4, where the truncated two-point correlation function for a class of lattice spin systems was proved to have exponential decayat low temperature, under a weak coupling assumption. In this paper we compute the asymptotics of the correlation function as the temperature goes to zero. This paper thus extends ref. 3 in two directions: The Hamiltonian function is allowed to have several local minima other than a unique global minimum, and we do not require translation invariance of the Hamiltonian function. We are in particular able to handle spin systems on a general lattice.