  J.J.L. Vel$\acute{a}$zquez in 1994 used the degree theory to show that there
is a perturbation of Simons' cone, starting from which the mean curvature flow
develops a type $\mathrm{II}$ singularity at the origin. He also showed that
under a proper time-dependent rescaling of the solution around the origin, the
rescaled flow converges in the $C^{0}$ sense to a minimal hypersurface which is
tangent to Simons' cone at infinity. In this paper, we prove that the rescaled
flow actually converges locally smoothly to the minimal hypersurface, which
appears to be the singularity model of the type $\mathrm{II}$ singularity. In
addition, we show that the mean curvature of the solution blows up near the
origin at a rate which is smaller than that of the second fundamental form.
