  We present a detailed study of the properties of the phase transition in the
four-dimensional compact U(1) lattice gauge theory supplemented by a monopole
term, for values of the monopole coupling $\lambda$ such that the transition is
of second order. By a finite size analysis we show that at $\lambda= 0.9$ the
critical exponent is already characteristic of a second-order transition.
Moreover, we find that this exponent is definitely different from the one of
the Gaussian case. We further observe that the monopole density becomes
approximately constant in the second-order region. Finally we reveal the
unexpected phenomenon that the phase transition persists up to very large
values of $\lambda$, where the transition moves to (large) negative $\beta$.
