We consider the numerical approximation of axisymmetric mean curvature flow
with the help of linear finite elements. In the case of a closed genus-1
surface, we derive optimal error bounds with respect to the $L^2$-- and
$H^1$--norms for a fully discrete approximation. We perform convergence
experiments to confirm the theoretical results, and also present numerical
simulations for some genus-0 and genus-1 surfaces.