  We provide the classification of eternal (or ancient) solutions of the
two-dimensional Ricci flow, which is equivalent to the fast diffusion equation
$ \frac{\partial u}{\partial t} = \Delta \log u $ on $ \R^2 \times \R.$ We show
that, under the necessary assumption that for every $t \in \R$, the solution
$u(\cdot, t)$ defines a complete metric of bounded curvature and bounded width,
$u$ is a gradient soliton of the form $ U(x,t) = \frac{2}{\beta (|x-x_0|^2 +
\delta e^{2\beta t})}$, for some $x_0 \in \R^2$ and some constants $\beta >0$
and $\delta >0$.
