Let $M$ be a compact smooth manifold, let $d: M \times M \to [0,\infty)$ be any metric inducing the topology of $M$, and let $f: M \to M$ be a $C^1$ Anosov diffeomorphism. Then $f$ is $C^1$ structurally stable: there exists a $C^1$ neighbourhood $\mathcal{U}$ of $f$ in $\operatorname{Diff}^1(M)$ such that, for every $g \in \mathcal{U}$, there exists a homeomorphism $h: M \to M$ satisfying

\begin{align*} h \circ f = g \circ h. \end{align*}

Moreover, for every neighbourhood $\mathcal{V}$ of $\operatorname{id}_M$ in $C^0(M,M)$, the neighbourhood $\mathcal{U}$ may be chosen so that $h \in \mathcal{V}$.