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In this paper we prove that every smooth complete closed complex hypersurfacein the open unit ball $\mathbb{B}_n$ of $\mathbb{C}^n$ $(n\ge 2)$ is a levelset of a noncritical holomorphic function on $\mathbb{B}_n$ all of whose levelsets are complete. This shows that $\mathbb{B}_n$ admits a nonsingularholomorphic foliation by smooth complete closed complex hypersurfaces and, whatis the main point, that every hypersurface in $\mathbb{B}_n$ of this type canbe embedded into such a foliation. We establish a more general result in whichneither completeness nor smoothness of the given hypersurface is required.  Furthermore, we obtain a similar result for complex submanifolds of arbitrarypositive codimension and prove the existence of a nonsingular holomorphicsubmersion foliation of $\mathbb{B}_n$ by smooth complete closed complexsubmanifolds of any pure codimension $q\in\{1,\ldots,n-1\}$.

Complete complex hypersurfaces in the ball come in foliations