  Tensor-scalar theory of gravity allows the generation of gravitational waves
from astrophysical sources, like Supernov\ae{}, even in the spherical case.
That motivated us to study the collapse of a degenerate stellar core, within
tensor-scalar gravity, leading to the formation of a neutron star through a
bounce and the formation of a shock. We discuss in this paper the effects of
the scalar field on the evolution of the system, as well as the appearance of
strong non-perturbative effects of this scalar field (the so-called
``spontaneous scalarization''). As a main result, we describe the resulting
gravitational monopolar radiation (form and amplitude) and discuss the
possibility of its detection by the gravitational detectors currently under
construction, taking into account the existing constraints on the scalar field.
  From the numerical point of view it is worthy to point out that we have
developed a combined code which uses pseudo-spectral methods, for the evolution
of the scalar field, and High Resolution Shock-Capturing schemes, for the
evolution of the hydrodynamical system. Although this code has been used to
integrate the field equations of that theory of gravity, in the spherically
symmetric case, a by-product of the present work is to gain experience for an
ulterior extension to multidimensional problems in Numerical Relativity of such
numerical strategy.
