A realistic pseudopotential model is introduced to investigate the phonon-induced spin relaxation of conduction electrons in bulk silicon. We find a surprisingly subtle interference of the Elliott and Yafet processes affecting the spin relaxation over a wide temperature range, suppressing the significance of the intravalley spin-flip scattering, previously considered dominant, above roughly 120 K. The calculated spin relaxation times T-1 agree with the spin resonance and spin injection data, following a T-3 temperature dependence. The valley anisotropy of T-1 and the spin relaxation rates for hot electrons are predicted.