In this paper we shall prove that the measure algebra M(G) of a locally compact group G is amenable as a Banach algebra if and only if G is discrete and amenable as a group. Our contribution is to resolve a conjecture by proving that M(G) is not amenable in the case where the group G is not discrete. Indeed, we shall prove a much stronger result: the measure algebra of a non-discrete, locally compact group has a non-zero, continuous point derivation at a certain character on the algebra.