A new method to compute fully differential double photoionization cross sections of atoms has been devised and fully developed for two-electron systems. The method exploits the Green function for two noninteracting electrons in the field of a nuclear charge to infer the effects of the residual potential projected on a set of L-2-basis functions. Test calculations oil helium at 100 eV excess energy indicate that, as long as the relevant part of the interaction potential is accounted for, the fully differential cross sections calculated in acceleration and velocity gauges converge in absolute value and reproduce measured angular distributions with a tunable accuracy. Generalization of the method to treat double photoionization of many-electron atoms is sketched.