  We consider the possibility to produce a bouncing universe in the framework
of scalar-tensor gravity models in which the scalar field potential may be
negative, and even unbounded from below. We find a set of viable solutions with
nonzero measure in the space of initial conditions passing a bounce, even in
the presence of a radiation component, and approaching a constant gravitational
coupling afterwards. Hence we have a model with a minimal modification of
gravity in order to produce a bounce in the early universe with gravity tending
dynamically to general relativity (GR) after the bounce.
