  We prove that, starting at an initial metric $g(0)=e^{2u_0}(dx^2+dy^2)$ on
$\mathbb{R}^2$ with bounded scalar curvature and bounded $u_0$, the Ricci flow
$\partial_t g(t)=-R_{g(t)}g(t)$ converges to a flat metric on $\mathbb{R}^2$.
