The time-dependent close-coupling method for the double photoionization of atoms is extended to compute fully differential nuclear-recoil cross sections. Excellent agreement is found between our calculated double-photoionization total cross sections of He and the measurements at all photon energies. Differential cross-section results are presented for the single-photon double ionization of He at photon energies of 99, 125, and 225 eV. The single-differential cross-section results at 99 eV agree with previous theory and experiment. Symmetric momentum distributions are found in the plane perpendicular to the polarization direction, while dipolelike momentum distributions are found in the other two planes. The variation of the nuclear-recoil triple-differential cross sections of He${}^{2+}$ with the nuclear-recoil momenta are presented. The total cross sections for the nuclear-recoil of He${}^{2+}$ match the corresponding ones for the double electron ejection of He for all the studied photon energies.