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This paper presents a solution to the problem of constructing Catmull-Clark subdivision surfaces interpolating given meshes of cubic B-spline curves. The solution is supplemented by an account of the continuity of the constructed surfaces at and around the extraordinary points of the corresponding meshes. This is conducted through an analysis of the corresponding subdivision matrix, which comes in parameterized form, thus adding a degree of freedom to the process. This degree of freedom gives room for manipulating the quality of the resulting surface without violating the initial interpolation constraints.