  We consider a geometrically fully nonlinear variational model for thin
elastic sheets that contain a single disclination. The free elastic energy
contains the thickness $h$ as a small parameter. We give an improvement of a
recently proved energy scaling law, removing the next-to leading order terms in
the lower bound. Then we prove the convergence of (almost-)minimizers of the
free elastic energy towards the shape of a radially symmetric cone, up to
Euclidean motions, weakly in the spaces $W^{2,2}(B_1\setminus
B_\rho;\mathbb{R}^3)$ for every $0<\rho<1$, as the thickness $h$ is sent to 0.
