  We study the Gauss map of minimal surfaces in the Heisenberg group
$\mathrm{Nil}_3$ endowed with a left-invariant Riemannian metric. We prove that
the Gauss map of a nowhere vertical minimal surface is harmonic into the
hyperbolic plane $\mathbb{H}^2$. Conversely, any nowhere antiholomorphic
harmonic map into $\mathbb{H}^2$ is the Gauss map of a nowhere vertical minimal
surface. Finally, we study the image of the Gauss map of complete nowhere
vertical minimal surfaces.
