We provide an exhaustive spectral analysis of the two-dimensional periodic
square graph lattice with a magnetic field. We show that the spectrum consists
of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum
of a certain discrete operator under the discriminant (Lyapunov function) of a
suitable Kronig-Penney Hamiltonian. In particular, between any two Dirichlet
eigenvalues the spectrum is a Cantor set for an irrational flux, and is
absolutely continuous and has a band structure for a rational flux. The
Dirichlet eigenvalues can be isolated or embedded, subject to the choice of
parameters. Conditions for both possibilities are given. We show that
generically there are infinitely many gaps in the spectrum, and the
Bethe-Sommerfeld conjecture fails in this case.