For $O$ a bounded domain in $Rd$ and a given smooth function $g:O\to R$, we consider the statistical nonlinear inverse problem of recovering the conductivity $f>0$ in the divergence form equation $$ \nabla\cdot(f\nabla u)=g\ \textrm{on}\ \mathcal{O}, \quad u=0\ \textrm{on}\ \partial\mathcal{O}, $$ from $N$ discrete noisy point evaluations of the solution $u=uf$ on $O$. We study the statistical performance of Bayesian nonparametric procedures based on a flexible class of Gaussian (or hierarchical Gaussian) process priors, whose implementation is feasible by MCMC methods. We show that, as the number $N$ of measurements increases, the resulting posterior distributions concentrate around the true parameter generating the data, and derive a convergence rate $N-\lambda ,\lambda >0,$ for the reconstruction error of the associated posterior means, in $L2(O)$-distance.