We show that spin-dependent electron-phonon interaction in the energy relaxation of a two-dimensional electron gas results in equal and oppositely directed currents in the spin-up and spin-down subbands yielding a pure spin current. In our experiments on SiGe heterostructures the pure spin current is converted into an electric current applying a magnetic field that lifts the cancellation of the two partial charge flows. A microscopic theory of this effect, taking account of the asymmetry of the relaxation process, is developed and is in good agreement with the experimental data.