In the paper, we consider a one-dimensional difference Schrödinger equation ψ(z+h)+ψ(z−h)+ λυ(z)ψ(z) = Eψ(z) with a periodic potential υ. In the case where the potential in a neighborhood of R is real analytic, as well as in the case where, the potential has a finite number of simple poles per period, for small values of the coupling constant λ, we describe the asymptotics of a monodromy matrix.