  We present a modified version of the one-dimensional sine-Gordon that
exhibits a thermodynamic, roughening phase transition, in analogy with the 2D
usual sine-Gordon model. The model is suited to study the crystalline growth
over an impenetrable substrate and to describe the wetting transition of a
liquid that forms layers. We use the transfer integral technique to write down
the pseudo-Schr\"odinger equation for the model, which allows to obtain some
analytical insight, and to compute numerically the free energy from the exact
transfer operator. We compare the results with Monte Carlo simulations of the
model, finding a perfect agreement between both procedures. We thus establish
that the model shows a phase transition between a low temperature flat phase
and a high temperature rough one. The fact that the model is one dimensional
and that it has a true phase transition makes it an ideal framework for further
studies of roughening phase transitions.
