  We consider the ground state properties of an inhomogeneous two-dimensional
Bose gas with a repulsive, short range pair interaction and an external
confining potential. In the limit when the particle number $N$ is large but
$\bar\rho a^2$ is small, where $\bar\rho$ is the average particle density and
$a$ the scattering length, the ground state energy and density are rigorously
shown to be given to leading order by a Gross-Pitaevskii (GP) energy functional
with a coupling constant $g\sim 1/|\ln(\bar\rho a^2)|$. In contrast to the 3D
case the coupling constant depends on $N$ through the mean density. The GP
energy per particle depends only on $Ng$. In 2D this parameter is typically so
large that the gradient term in the GP energy functional is negligible and the
simpler description by a Thomas-Fermi type functional is adequate.
