  In this paper we consider a discrete Klein-Gordon (dKG) equation on $\ZZ^d$
in the limit of the discrete nonlinear Schrodinger (dNLS) equation, for which
small-amplitude breathers have precise scaling with respect to the small
coupling strength $\eps$. By using the classical Lyapunov-Schmidt method, we
show existence and linear stability of the KG breather from existence and
linear stability of the corresponding dNLS soliton. Nonlinear stability, for an
exponentially long time scale of the order $\mathcal{O}(\exp(\eps^{-1}))$, is
also obtained via the normal form technique, together with higher order
approximations of the KG breather through perturbations of the corresponding
dNLS soliton.
