Let G be a permutation group acting on a set <br/>. A subset of <br/> is a base for G if<br/>its pointwise stabilizer in G is trivial. We write b(G) for the minimal size of a base for<br/>G. We determine the precise value of b(G) for every primitive almost simple sporadic<br/>group G, with the exception of two cases involving the Baby Monster group. As a<br/>corollary, we deduce that b(G) 6 7, with equality if and only if G is the Mathieu group<br/>M24 in its natural action on 24 points. This settles a conjecture of Cameron
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