  We study the Ginzburg-Landau equations in order to describe a two-dimensional
superconductor in a bounded domain. Using the properties of a particular
integrability point ($\kappa = 1/ \sqrt2$) of these nonlinear equations which
allows vortex solutions, we obtain a closed expression for the energy of the
superconductor. The presence of the boundary provides a selection mechanism for
the number of vortices.
  A perturbation analysis around $\kappa = 1/ \sqrt2$ enables us to include the
effects of the vortex interactions and to describe quantitatively the
magnetization curves recently measured on small superconducting disks.
  We also calculate the optimal vortex configuration and obtain an expression
for the confining potential away from the London limit.
