  This paper is concerned with steady vortex rings in an ideal fluid of uniform
density, which are special global axi-symmetric solutions of the
three-dimensional incompressible Euler equation. We systematically establish
the existence, uniqueness and nonlinear orbital stability of steady vortex
rings of small cross-section for which the potential vorticity is constant
throughout the core. The latter two answer a long-standing question since the
pioneering work of Fraenkel and Berger \cite{BF1} (Acta Math., 1974). To
achieve our goal, we rescale the Stokes stream function of vortex ring by its
cross-section radius, and expand it at the well-known Rankine vortex using
Taylor's formula, where the estimates for coefficients are obtained by a
decomposition according to Green's function and local Pohozaev identities. The
main observations are: The stream function is even and has a translational
invariance in $z$-direction; the zero point for the $r$-coefficient of linear
term in its expansion determines the asymptotic location of vortex ring, which
appears as the condition to eliminate the degenerate direction in
Lyapunov-Schmidt reduction argument for existence; the non-vanishing condition
of the second order $r$-coefficient at foresaid zero point is verified as one
of the essential factors for uniqueness, while the negativity means that these
vortex rings maximizes the functional composed of kinetic energy and impulse.
By applying the Arnol'd's dual variational principle together with the
uniqueness result, we are finally able to prove the nonlinear orbital stability
of thin vortex rings. This result gives a large class of stable vortex rings
supported on topological tori, which is different from Hill's spherical vortex
discussed by Choi \cite{Choi20} (Comm. Pure Appl. Math., 2023).
