Let $(X,\omega)$ be a compact Kähler manifold and let $V\subset X$ be a positive-dimensional compact complex submanifold. Let $T$ be a Kähler current with analytic singularities on $V$ in the Kähler class $[\omega|_{V}]$. Then there exists a Kähler current $\tilde{T}$ on $X$ in the class $[\omega]$ with $T=\tilde{T}|_{V}$.