  The order of the Coulomb-Higgs transition in the U(1)-Higgs model with
unfrozen modulus of the scalar field is studied. Large lattices (up to $24^4$
in one case) and high statistics are used. We fix $\beta =1.15$ and explore
specially a region of $\lambda$-values where metastability is observed. We
study the thermodynamical limit of several observables, in particular, the
latent heat, the specific heat, the decrement of the free energy between the
maxima and the central minimum of the two-peaked histogram, the Binder cumulant
and the displacement of the critical coupling with the lattice size. The
results point towards a second order transition for $\lambda \gtrsim 0.005$,
while for smaller values of $\lambda$ the strong metastability growing with the
lattice size seems to derive from a first order character.
