  We study long-time existence and asymptotic behavior for the $L^2$-gradient
flow of the Willmore energy, under the condition that the initial datum is a
torus of revolution. We show that if an initial datum has Willmore energy below
$8\pi$ then the solution of the Willmore flow converges for $t \rightarrow
\infty$ to the Clifford Torus, possibly rescaled and translated. The energy
threshold of $8\pi$ turns out to be optimal for such a convergence result. We
give an application to the conformally constrained Willmore minimization
problem.
