We construct interpolatory cubature rules on the two-dimensional sphere, using the fundamental system of points obtained by Laín Fernández [Polynomial Bases on the Sphere, Logos-Verlag, Berlin, 2003; Localized polynomial bases on the sphere, Electron. Trans. Numer. Anal. 19 (2005) 84-93]. The weights of the cubature rules are calculated explicitly. We also discuss the cases when this cubature leads to positive weights. Finally, we study the possibility to construct spherical designs and the degree of exactness.