Let \(C\) be a smooth complex curve of genus three. By Andreotti's proof of Torelli's theorem the curve \(C\) is determined by the Gauss map \(\gamma\) of the theta divisor \(\Theta\) associated to \(C\). The \((-1)\)-map on the Jacobian of \(C\) restricts to a \(\mathbb{Z}/2\mathbb{Z}\)-action on \(\Theta\) with respect to which \(\gamma\) is invariant. It is shown that for nonhyperelliptic curves \(C\) the Gauss map is locally \(\mathbb{Z}/2\mathbb{Z}\)- stable if and only if \(C\) admits only normal Weierstrass points, and for hyperelliptic \(C\) \(\gamma\) is always locally \(\mathbb{Z}/2\mathbb{Z}\)-stable. From this it follows in particular that the set of isomorphism classes of smooth nonhyperelliptic genus-three curves with locally \(\mathbb{Z}/2\mathbb{Z}\)- stable Gauss map is a nonempty Zariski open subsets of \({\mathcal M}_ 3\).