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 $ u$ is close to, however, different from the gaussian value and that in the range considered $ u$ increases somewhat with $lambda$.
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