Let X be a smooth projective algebraic curve of genus > 1 over and algebraically closed field k of characteristic p > 0. Denote by Bunn (resp. Locn) the moduli stack of vector bundles of rank n on X (resp. the moduli stack of vector bundles of rank n endowed with a connection). Let also DBUnn denote the sheaf of crystalline differential operators on Bunn (cf. e.g. ). In this paper we construct an equivalence φn between the bounded derived category Db(M(OLocnO)) of quasi-coherent sheaves on some open subset Locn O ⊂ Locn and the bounded derived category Db(M(DBunnO)) of the category of modules over some localization DBunnO of DBunn. We show that this equivalence satisfies the Hecke eigen-value property in the manner predicted by the geometric Langlands conjecture. In particular, for any ε ∈ Locn O we construct a "Hecke eigen-module" Autε. The main tools used in the construction are the Azumaya property of DBunn (cf. ) and the geometry of the Hitchin integrable system. The functor φn is defined via a twisted version of the Fourier-Mukai transform.