  The non-existence of asymptotically flat, neutral black holes and
asymptotically flat, charged black holes in the Maxwell electrodynamics, with
non-trivial scalar field has been proved for a large class of scalar-tensor
theories. The no-scalar-hair theorems, however, do not apply in the case of
non-linear electrodynamics. In the present work numerical solutions describing
charged black holes coupled to Born-Infeld type non-linear electrodynamics in
scalar-tensor theories of gravity with massless scalar field are found. The
causal structure and properties of the solutions are studied, and a comparison
between these solutions and the corresponding solutions in the General
Relativity is made. The presence of the scalar field leads to a much more
simple causal structure. The present class of black holes has a single,
non-degenerate horizon, i.e., its causal structure resembles that of the
Schwarzschild black hole.
