The Erdős–Simonovits stability theorem states that for all ε > 0 there exists α > 0 such that if G is a Kr+1-free graph on n vertices with e(G) > ex(n, Kr+1) − αn2, then one can remove εn2 edges from G to obtain an r-partite graph. Fu¨redi gave a short proof that one can choose α = ε. We give a bound for the relationship of α and ε which is asymptotically sharp as ε → 0.