  We consider an $N \times N$ random symmetric Toeplitz matrix with an i.i.d.
input sequence drawn from a distribution that lies in the domain of attraction
of an $\alpha$-stable law for $0 < \alpha < 2$. We show that under an
appropriate scaling, its empirical eigenvalue distribution, as $N \to \infty$,
converges weakly to a random symmetric probability distribution on
$\mathbb{R}$, which can be described as the expected spectral measure of a
certain random unbounded self-adjoint operator on $\ell^2(\mathbb{Z})$. The
limiting distribution turns out to be almost surely subgaussian. Furthermore,
the support of the limiting distribution is bounded almost surely if $0<\alpha
<1$ and is unbounded almost surely if $1\leq \alpha <2$.
