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We show that if $X$ and $Y$ are uniformly locally finite metric spaces whoseuniform 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 toMorita equivalence between $\cstu(X)$ and $\cstu(Y)$. As an application, weobtain that if $\Gamma$ and $\Lambda$ are finitely generated groups, then thecrossed 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.

Uniform Roe algebras of uniformly locally finite metric spaces are rigid