Let (X, d) be a compact metric space and let M(X ) denote the space of all ﬁnite signed Borel measures on X . Deﬁne I : M(X ) → R by I (µ) = X X d(x, y ) dµ(x) dµ(y ), and set M (X ) = sup I (µ), where µ ranges over the collection of signed measures in M(X ) of total mass 1. This paper, with an earlier and a subsequent paper [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric spaces. I and III], investigates the geometric constant M (X ) and its relationship to the metric properties of X and the functional-analytic properties of a certain subspace of M(X ) when equipped with a natural semi-inner product. Using the work of the earlier paper, this paper explores measures which attain the supremum deﬁning M (X ), sequences of measures which approximate the supremum when the supremum is not attained and conditions implying or equivalent to the ﬁniteness of M (X ).