A new method for constructing a smooth surface that interpolates the vertices of an arbitrary mesh is presented. The mesh can be open or closed. Normals specified at vertices of the mesh can also be interpolated. The interpolating surface is obtained by locally adjusting the limit surface of the given mesh (viewed as the control mesh of a Catmull-Clark subdivision surface) so that the modified surface would interpolate all the vertices of the given mesh. The local adjustment process is achieved through locally blending the limit surface with a surface defined by non-uniform transformations of the limit surface. This local blending process can also be used to smooth out the shape of the interpolating surface. Hence, a surface fairing process is not needed in the new method. Because the interpolation process does not require solving a system of linear equations, the method can handle meshes with large number of vertices. Test results show that the new method leads to good interpolation results even for complicated data sets. The new method is demonstrated with the Catmull-Clark subdivision scheme. But with some minor modification, one should be albe to apply this method to other parametrizable subdivision schemes as well.