  We study Glauber dynamics for the Ising model on the complete graph on $n$
vertices, known as the Curie-Weiss Model. It is well known that at high
temperature ($\beta < 1$) the mixing time is $\Theta(n\log n)$, whereas at low
temperature ($\beta > 1$) it is $\exp(\Theta(n))$. Recently, Levin, Luczak and
Peres considered a censored version of this dynamics, which is restricted to
non-negative magnetization. They proved that for fixed $\beta > 1$, the
mixing-time of this model is $\Theta(n\log n)$, analogous to the
high-temperature regime of the original dynamics. Furthermore, they showed
\emph{cutoff} for the original dynamics for fixed $\beta<1$. The question
whether the censored dynamics also exhibits cutoff remained unsettled.
  In a companion paper, we extended the results of Levin et al. into a complete
characterization of the mixing-time for the Currie-Weiss model. Namely, we
found a scaling window of order $1/\sqrt{n}$ around the critical temperature
$\beta_c=1$, beyond which there is cutoff at high temperature. However,
determining the behavior of the censored dynamics outside this critical window
seemed significantly more challenging.
  In this work we answer the above question in the affirmative, and establish
the cutoff point and its window for the censored dynamics beyond the critical
window, thus completing its analogy to the original dynamics at high
temperature. Namely, if $\beta = 1 + \delta$ for some $\delta > 0$ with
$\delta^2 n \to \infty$, then the mixing-time has order $(n /
\delta)\log(\delta^2 n)$. The cutoff constant is $(1/2+[2(\zeta^2 \beta /
\delta - 1)]^{-1})$, where $\zeta$ is the unique positive root of
$g(x)=\tanh(\beta x)-x$, and the cutoff window has order $n / \delta$.
