We report on the optical generation of a pure spin current at the indirect gap of bulk Si at room temperature in the photon energy range comprised between 1.2 and 1.8 eV. Spin-polarized electrons are promoted to the Δ-valleys of the Si Brillouin zone by circularly polarized light. The photo-generated spin current is then detected by exploiting a Schottky Pt/Si(001) junction: spin-polarized electrons diffuse toward the Pt/Si interface and enter the Pt layer where the spin current is converted into a transverse electromotive field through the inverse spin-Hall effect (ISHE). The photon energy dependence of the ISHE signal is interpreted in the frame of a one-dimensional spin drift-diffusion model, which allows estimating the electron spin lifetime to be τs=15±5 ns.