In this work, we consider the model of $\mathbb{ S}^2\times\mathbb{ R}$ isometric to $\mathbb{ R}^3\setminus \{ 0\}$, endowed with a metric conformally equivalent to the Euclidean metric of $\mathbb{ R}^3$, and we define a Gauss map for surfaces in this model likewise in the $3-$Euclidean space. We show as a main result that any two minimal conformal immersions in $\mathbb{ S}^2\times\mathbb{ R}$ with the same non-constant Gauss map differ by only two types of ambient isometries: either $f=(\mathrm{ id},T)$, where $T$ is a translation on $\mathbb{ R}$, or $f=(\mathcal{ A},T)$, where $\mathcal{ A}$ denotes the antipodal map on $\mathbb{ S}^2$. Moreover, if the Gauss map is singular, we show that it is necessarily constant, and then only vertical cylinders over geodesics of $\mathbb{ S}^2$ in $\mathbb{ S}^2\times\mathbb{ R}$ appear with this assumption. We also study some particular cases, among them we prove that there is no minimal conformal immersion in $\mathbb{ S}^2\times\mathbb{ R}$ which the Gauss map is a non-constant anti-holomorphic map.
Categories: math.DG