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We report the first parameter-free calculations of self-diffusion constants in silicon. We have computed diffusion constants for the defect-mediated mechanisms using the local-density approximation in combination with ab initio molecular-dynamics simulations and obtained the diffusion constant for the concerted exchange mechanism from earlier results by Pandey and Kaxiras. We obtain diffusion constants in the range of the experimental values, with the self-interstitial mechanism dominating over the contribution of the other mechanisms.