  We consider an embedded convex ancient solution $\Gamma_t$ to the curve
shortening flow in $\mathbb{R}^2$. We prove that there are only two
possibilities: the family $\Gamma_t$ is either the family of contracting
circles, which is a type I ancient solution, or the family of evolving Angenent
ovals, which correspond to a type II ancient solution to the curve shortening
flow. We also give a necessary and sufficient curvature condition for an
embedded, closed ancient solution to the curve shortening flow to be convex.
