  We present results from a numerical simulation of the two-dimensional
Euclidean Wess-Zumino model. In the continuum the theory possesses N=1
supersymmetry. The lattice model we employ was analyzed by Golterman and
Petcher in \cite{susy} where a perturbative proof was given that the continuum
supersymmetric Ward identities are recovered without finite tuning in the limit
of vanishing lattice spacing. Our simulations demonstrate the existence of
important non-perturbative effects in finite volumes which modify these
conclusions. It appears that in certain regions of parameter space the vacuum
state can contain solitons corresponding to field configurations which
interpolate between different classical vacua. In the background of these
solitons supersymmetry is partially broken and a light fermion mode is
observed. At fixed coupling the critical mass separating phases of broken and
unbroken supersymmetry appears to be volume dependent. We discuss the
implications of our results for continuum supersymmetry breaking.
