  Using the monomer--dimer representation of the lattice Schwinger model, with
$N_f =1$ Wilson fermions in the strong--coupling regime ($\beta=0$), we
evaluate its partition function, $Z$, exactly on finite lattices. By studying
the zeroes of $Z(k)$ in the complex plane $(Re(k),Im(k))$ for a large number of
small lattices, we find the zeroes closest to the real axis for infinite
stripes in temporal direction and spatial extent $S=2$ and 3. We find evidence
for the existence of a critical value for the hopping parameter in the
thermodynamic limit $S\rightarrow \infty$ on the real axis at about $k_c \simeq
0.39$. By looking at the behaviour of quantities, such as the chiral
condensate, the chiral susceptibility and the third derivative of $Z$ with
respect to $1/2k$, close to the critical point $k_c$, we find some indications
for a continuous phase transition.
