  Let $X$ and $Y$ be Polish spaces with non-atomic Borel measures $\mu$ and
$\nu$ of full support. Suppose that $T$ and $S$ are ergodic non-singular
homeomorphisms of $(X,\mu)$ and $(Y,\nu)$ with continuous Radon-Nikodym
derivatives. Suppose that either they are both of type $III_1$ or that they are
both of type $III_\lambda$, $0<\lambda<1$ and, in the $III_\lambda$ case,
suppose in addition that both `topological asymptotic ranges' (defined in the
article) are $\log\lambda\cdot\Bbb Z$. Then there exist invariant dense
$G_\delta$-subsets $X'\subset X$ and $Y'\subset Y$ of full measure and a
non-singular homeomorphism $\phi: X' \to Y'$ which is an orbit equivalence
between $T|_{X'}$ and $S|_{Y'}$, that is $\phi\{T^{i}x\} = \{S^{i}x\}$ for all
$x \in X'$. Moreover the Radon-Nikodym derivative $d\nu\circ\phi/d\mu$ is
continuous on $X'$ and, letting $S' = \phi^{-1}S \phi$ we have $Tx=
{S'}^{n(x)}x$ and $S' = T^{m(x)}x$ where $n$ and $m$ are continuous on $X'$.
