  We study the convergence of an axially symmetric hypersurface evolving by
volume preserving mean curvature flow. Assuming the surface is not pinching off
along the axis at any time during the flow, and without any additional
conditions, as for example on the curvature, we prove that it converges to a
hemisphere, when the hypersurface has a free boundary and satisfies Neumann
boundary data, and to a sphere when it is compact without boundary.
