  We consider neutron star configurations in the scalar-tensor theory of
gravity with the coupling between the kinetic term of a scalar field and the
Einstein tensor (such the model is a subclass of Horndeski gravity). Neutron
stars in this model were studied earlier for the special case with a vanishing
``bare'' cosmological constant, $\Lambda_0=0$, and a vanishing standard kinetic
term, $\alpha=0$. This special case is of interest because it admits so-called
stealth configuration, i.e. vacuum configuration with nontrivial scalar field
and the Schwarzschild metric. However, generally one has $\Lambda_0\not=0$ and
$\alpha\not=0$ and in this case a vacuum configuration is represented as an
asymptotically anti-de Sitter (AdS) black hole solution with the nontrivial
scalar field. We construct neutron star configurations in this general case and
show that resulting diagrams describing the relation between mass and radius of
the star essentially differ from those obtained in GR or the particular model
with $\alpha=\Lambda_0=0$. Instead, the mass-radius diagrams are similar to
those obtained for so-called bare strange stars when a star radius decreases
monotonically with decreasing mass. We show also that neutron stars in the
theory of gravity with nonminimal derivative coupling are more compact
comparing to those in GR or the particular model with $\alpha=\Lambda_0=0$ and
suggest a way to estimate possible values of the parameter of nonminimal
coupling $\ell$. {\color{red} At last, using the Regge-Wheeler method, we
discuss briefly the stability of obtained neutron star configurations.
