Abstract. Let H be a weak Hopf algebra and let A be an H-comodule algebra with subalgebra of coinvariants A H . In this paper we introduce the notion of H-Galois extension with normal basis and we prove that A H → A is an H-Galois extension with normal basis if and only if A H → A is an H-cleft extension which admits a convolution invertible total integral. As a consequence, if H is cocommutative and A commutative, we obtain a bijective correspondence between the second cohomology group H 2