  Given a resistor network on $\mathbb Z^d$ with nearest-neighbor conductances,
the effective conductance in a finite set with a given boundary condition is
the the minimum of the Dirichlet energy over functions with the prescribed
boundary values. For shift-ergodic conductances, linear (Dirichlet) boundary
conditions and square boxes, the effective conductance scaled by the volume of
the box converges to a deterministic limit as the box-size tends to infinity.
Here we prove that, for i.i.d. conductances with a small ellipticity contrast,
also a (non-degenerate) central limit theorem holds. The proof is based on the
corrector method and the Martingale Central Limit Theorem; a key integrability
condition is furnished by the Meyers estimate. More general domains, boundary
conditions and ellipticity contrasts will be addressed in a subsequent paper.
