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In this paper we present the general theory of cleft extensions for acocommutative weak Hopf algebra $H$. For a weak left $H$-module algebra weobtain a bijective correspondence between the isomorphisms classes of $H$-cleftextensions $A_{H}\hookrightarrow A$, where $A_{H}$ is the subalgebra ofcoinvariants, and the equivalence classes of crossed systems for $H$ over$A_H$. Finally, we establish a bijection between the set of equivalence classesof crossed systems with a fixed weak $H$-module algebra structure and thesecond cohomology group $H_{\varphi_{Z(A_H)}}^2(H, Z(A_H))$, where $Z(A_H)$ isthe center of $A_H$.

Integrals and crossed products over weak Hopf algebras