Given two von Neumann algebras M and N acting on the same Hilbert space, d(M;N) is defined to be the Hausdor distance between their unit balls. The Kadison-Kastler problem asks whether two sufficiently close von Neumann algebras are spatially isomorphic. In this article, we show that if P0 is an injective von Neumann algebra with a cyclic tracial vector, G is a free group, ? is a free action of G on P0 and N is a von Neumann algebra such that d(N; P0 x| ? G) < 1/7 o 10^-7, then N and P0 x| ? G are spatially isomorphic. Suitable choices of the actions give the first examples of infinite noninjective factors for which this problem has a positive solution.