Let Ω be a smooth nowhere-zero n-form on a non-compact n-dimensional manifold Y. We study the homology of the group DiffΩY, consisting of all diffeomorphisms of Y preserving Ω, provided with the discrete topology. In particular, if Ω is the standard volume form dx1∧ … ∧ dxn on Rn, and if n¦ 2,3, we show that the natural maps B D i f f Ω R n ↪ B E m b Ω R n ⟶ ≃ B Γ Ω n induce isomorphisms of integral homology. Here EmbΩRn is the discrete monoid of smooth embeddings of Rn preserving Ω, and BTΩn is the Haefliger classifying space for codimension-n smooth foliations with a smooth non-vanishing transverse closed n-form. (Analogous statements for the group of all diffeomorphisms of Rn were proved by Segal in [16] with no restriction on dimension.) It follows that if n ¦ 2,3 the natural map BDiff ΩRn → B SO(n), where SO(n) is given its usual topology, induces an isomorphism on integral homology Hi for 0 ⩽ i < n, and a surjection on Hn with kernel isomorphic to R. If Y is a non-compact manifold which is diffeomorphic to the interior of a compact n-dimensional manifold with boundary, and if every end of Y has infinite -volume, we show that H 1 ( B D i f f Ω 0 Y ; Z ) ≅ H n - 1 ( Y ; R ) , providing that n¦ 2,3. Here Diffο0 ⊂ Diffο denotes the subgroup consisting of elements smoothly isotopic to the identity.