  We present a mathematically rigorous analysis of the ground state of a
dilute, interacting Bose gas in a three-dimensional trap that is strongly
confining in one direction so that the system becomes effectively
two-dimensional. The parameters involved are the particle number, $N\gg 1$, the
two-dimensional extension, $\bar L$, of the gas cloud in the trap, the
thickness, $h\ll \bar L$ of the trap, and the scattering length $a$ of the
interaction potential. Our analysis starts from the full many-body Hamiltonian
with an interaction potential that is assumed to be repulsive, radially
symmetric and of short range, but otherwise arbitrary. In particular, hard
cores are allowed. Under the premisses that the confining energy, $\sim 1/h^2$,
is much larger than the internal energy per particle, and $a/h\to 0$, we prove
that the system can be treated as a gas of two-dimensional bosons with
scattering length $a_{\rm 2D}= h\exp(-(\hbox{\rm const.)}h/a)$. In the
parameter region where $a/h\ll |\ln(\bar\rho h^2)|^{-1}$, with $\bar\rho\sim
N/\bar L^2$ the mean density, the system is described by a two-dimensional
Gross-Pitaevskii density functional with coupling parameter $\sim Na/h$. If
$|\ln(\bar\rho h^2)|^{-1}\lesssim a/h$ the coupling parameter is $\sim N
|\ln(\bar\rho h^2)|^{-1}$ and thus independent of $a$. In both cases
Bose-Einstein condensation in the ground state holds, provided the coupling
parameter stays bounded.
