A smooth cube manifold M is a smooth n-manifold M together with a smooth cubification on M. The cube structure provides rays that are normal to the open k-subcubes. Using these rays we can construct "normal charts" in an obvious and natural way. A complete set of normal charts gives a (topological) "normal atlas" on M. If this atlas is smooth it is called a "normal smooth atlas" on M and induces a "normal smooth structure" on M (normal with respect to the cube structure). We prove that every smooth cube manifold has a normal smooth structure, which is diffeomorphic to the original one. This result also holds for smooth all-right-spherical manifolds.