  Born-Infeld black holes in the Scalar-Tensor Theories of Gravity, in the case
of massless scalar field, have been recently obtained. The aim of the current
paper is to study the effect from the inclusion of a potential for the scalar
field in the theory, through a combination of analytical techniques and
numerical methods. The black holes coupled to a massive scalar field have
richer causal structure in comparison to the massless scalar field case. In the
latter case, the black holes may have a second, inner horizon. The presence of
potential for the scalar field allows the existence of extremal black holes for
certain values of the mass of the scalar field and the magnetic (electric)
charge of the black hole. The linear stability against spherically symmetric
perturbations is studied. Arguments in favor of the general stability of the
solutions coming from the application of the "turning point" method are also
presented.
