We investigate the regularity of bounded weak solutions of scalar conservation laws with uniformly convex flux in space dimension one, satisfying an entropy condition with entropy production term that is a signed Radon measure. We prove that all such solutions belong to the Besov space B1/3,3_{∞,{ loc}} . Since C. de Lellis and M. Westdickenberg [11] have proved the existence of such solutions that do not belong to Bs,p_{q,{ loc}} if either s>1/max(p,3) or s=1/3 and 1≤ q or s=1/p with p≥ 3 and q<∞ , this regularizing effect is optimal. The proof is based on the kinetic formulation of scalar conservation laws and on an interaction estimate in physical space.
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