Abstract: A problem of homogenization of a divergence type second order uniformly elliptic operator is considered with arbitrary bounded rapidly oscillating periodic coefficients, either with periodic .outer. boundary conditions or in the whole space. It is proved that if the right-hand side is Gevrey regular (in particular analytic) then by optimally truncating the full two-scale asymptotic expansion for the solution one obtains an approximation with an exponentially small error. The optimality of the exponential error bound is established for a one-dimensional example by proving the analogous lower bound.