  The morphology of galaxy clusters reflects the epoch at which they formed and
hence depends on the value of the mean cosmological density, Omega. Recent
studies have shown that the distribution of dark matter in clusters can be
mapped from analysis of the small distortions in the shapes of background
galaxies induced by weak gravitational lensing in the cluster potential. We
construct new statistics to quantify the morphology of clusters which are
insensitive to limitations in the mass reconstruction procedure. By simulating
weak gravitational lensing in artificial clusters grown in numerical
simulations of the formation of clusters in three different cosmologies, we
obtain distributions of a quadrupole statistic which measures global deviations
from spherical symmetry in a cluster. These distributions are very sensitive to
the value of Omega_0 and, as a result, lensing observations of a small number
of clusters should be sufficient to place broad constraints on Omega_{0} and
certainly to distinguish between the extreme values of 0.2 and 1.
