A new quadrature rule for functions defined on the sphere is introduced. The nodes are defined as the points of the Cubed Sphere. The associated weights are defined in analogy to the trapezoidal rule on each panel of the Cubed Sphere. The formula enjoys a symmetry property ensuring that a proportion of 7/8 of all Spherical Harmonics is integrated exactly. Based on the remaining Spherical Harmonics, it is possible to define modified weights giving an enhanced quadrature rule. Numerical results show that the new quadrature is competitive with classical rules of the litterature. This second quadrature rule is believed to be of interest for applied mathematicians, physicists and engineers dealing with data located at the points of the Cubed Sphere.

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