  We study ground-state properties of the two-dimensional random-bond Ising
model with couplings having a concentration $p\in[0,1]$ of antiferromagnetic
and $(1-p)$ of ferromagnetic bonds. We apply an exact matching algorithm which
enables us the study of systems with linear dimension $L$ up to 700. We study
the behavior of the domain-wall energies and of the magnetization. We find that
the paramagnet-ferromagnet transition occurs at $p_c \sim 0.103$ compared to
the concentration $p_n\sim 0.109$ at the Nishimory point, which means that the
phase diagram of the model exhibits a reentrance. Furthermore, we find no
indications for an (intermediate) spin-glass ordering at finite temperature.
