  We investigate critical 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. It has been previously shown that at $\lambda= 0.9$ the
critical exponent is already characteristic of a second-order transition and
that it is different from the one of the Gaussian case. In the present study we
perform a finite size analysis at $\lambda=1.1$ to get information wether the
value of this exponent is universal.
