  We investigate the uniqueness of symmetric weak solutions to the stationary
Navier-Stokes equation in a two-dimensional exterior domain $\Omega$. It is
known that, under suitable symmetry condition on the domain and the data, the
problem admits at least one symmetric weak solution tending to zero at
infinity. Given two symmetric weak solutions $u$ and $v$, we show that if $u$
satisfies the energy inequality $\| \nabla u \|_{L^2 (\Omega)}^2 \le (f,u)$ and
$\sup_{x \in \Omega} (|x|+1)|v(x)|$ is sufficiently small, then $u=v$. The
proof relies upon a density property for the solenoidal vector field and the
Hardy inequality for symmetric functions.
