  We consider the quark-meson model with two quark flavors in a constant
external magnetic field $B$ at finite temperature $T$ and finite baryon
chemical potential $\mu_B$. We calculate the full renormalized effective
potential to one-loop order in perturbation theory. We study the system in the
large-$N_c$ limit, where we treat the bosonic modes at tree level. It is shown
that the system exhibits dynamical chiral symmetry breaking, i. e. that an
arbitrarily weak magnetic field breaks chiral symmetry dynamically, in
agreement with earlier calculations using the NJL model. We study the influence
on the phase transition of the fermionic vacuum fluctuations. For strong
magnetic fields, $|qB|\sim5m_{\pi}^2$ and in the chiral limit, the transition
is first order in the entire $\mu_B-T$ plane if vacuum fluctuations are not
included and second order if they are included. At the physical point, the
transition is a crossover for $\mu_B=0$ with and without vacuum fluctuations.
