  We study a two-dimensional rotating Bose-Einstein condensate confined by an
anharmonic trap in the framework of the Gross-Pitaevksii theory. We consider a
rapid rotation regime close to the transition to a giant vortex state. It was
proven in [M. Correggi {\it et al}, {\it J. Math. Phys. \textbf{53}(2012)] that
such a transition occurs when the angular velocity is of order $ \varepsilon
^{-4}$, with $ \varepsilon ^{-2} $ denoting the coefficient of the nonlinear
term in the Gross-Pitaevskii functional and $ \varepsilon \ll 1 $ (Thomas-Fermi
regime). In this paper we identify a finite value $ \Omega_{\mathrm{c}} $ such
that, if $ \Omega = \Omega_0/\varepsilon ^4 $ with $ \Omega_0 >
\Omega_{\mathrm{c}} $, the condensate is in the giant vortex phase. Under the
same condition we prove a refined energy asymptotics and an estimate of the
winding number of any Gross-Pitaevskii minimizer.
