An explicit global and unique isometric embedding into hyperbolic 3-space, ${H}^{3}$, of an axi-symmetric 2-surface with Gaussian curvature bounded below is given. In particular, this allows the embedding into ${H}^{3}$ of surfaces of revolution having negative, but finite, Gaussian curvature at smooth fixed points of the $U(1)$ isometry. As an example, we exhibit the global embedding of the Kerr-Newman event horizon into ${H}^{3}$, for arbitrary values of the angular momentum. For this example, considering a quotient of ${H}^{3}$ by the Picard group, we show that the hyperbolic embedding fits in a fundamental domain of the group up to a slightly larger value of the angular momentum than the limit for which a global embedding into Euclidean 3-space is possible. An embedding of the double-Kerr event horizon is also presented, as an example of an embedding that cannot be made global.