  We present a uniqueness theorem for the reduced C*-algebra of a twist
$\mathcal{E}$ over a Hausdorff \'etale groupoid $\mathcal{G}$. We show that the
interior $\mathcal{I}^\mathcal{E}$ of the isotropy of $\mathcal{E}$ is a twist
over the interior $\mathcal{I}^\mathcal{G}$ of the isotropy of $\mathcal{G}$,
and that the reduced twisted groupoid C*-algebra
$C_r^*(\mathcal{I}^\mathcal{G}; \mathcal{I}^\mathcal{E})$ embeds in
$C_r^*(\mathcal{G}; \mathcal{E})$. We also investigate the full and reduced
twisted C*-algebras of the isotropy groups of $\mathcal{G}$, and we provide a
sufficient condition under which states of (not necessarily unital) C*-algebras
have unique state extensions. We use these results to prove our uniqueness
theorem, which states that a C*-homomorphism of $C_r^*(\mathcal{G};
\mathcal{E})$ is injective if and only if its restriction to
$C_r^*(\mathcal{I}^\mathcal{G}; \mathcal{I}^\mathcal{E})$ is injective.
