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 $��$, 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 $��$ where the confinement-Coulomb phase transition turns out to become of second order. Performing a finite-size analysis we find that the critical exponent $��$ is close to, however, different from the gaussian value and that in the range considered $��$ increases somewhat with $��$.

15 pages, LaTeX, 3 figures