  We show that if $X$ and $Y$ are uniformly locally finite metric spaces whose
uniform Roe algebras, $\cstu(X)$ and $\cstu(Y)$, are isomorphic as
\cstar-algebras, then $X$ and $Y$ are coarsely equivalent metric spaces.
Moreover, we show that coarse equivalence between $X$ and $Y$ is equivalent to
Morita equivalence between $\cstu(X)$ and $\cstu(Y)$. As an application, we
obtain that if $\Gamma$ and $\Lambda$ are finitely generated groups, then the
crossed products $\ell_\infty(\Gamma)\rtimes_r\Gamma$ and $
\ell_\infty(\Lambda)\rtimes_r\Lambda$ are isomorphic if and only if $\Gamma$
and $\Lambda$ are bi-Lipschitz equivalent.
