  We investigate four-dimensional compact U(1) lattice gauge theory with a
monopole term added to the Wilson action. First we consider the phase structure
at negative $\beta$, revealing some properties of a third phase region there,
in particular the existence of a number of different states. Then our present
studies concentrate on larger values of the monopole coupling $\lambda$ where
the confinement-Coulomb phase transition turns out to become of second order.
Performing a finite-size analysis we find that the critical exponent $\nu$ is
close to, however, different from the gaussian value and that in the range
considered $\nu$ increases somewhat with $\lambda$.
