  The minimal Starobinsky supergravity with the inflaton (scalaron) and the
goldstino in a massive vector supermultiplet is coupled to the dilaton-axion
chiral superfield with the no-scale K\"ahler potential and a superpotential.
The Kachru-Kallosh-Linde-Trivedi (KKLT)-type mechanism in the presence of a
constant term in the superpotential is applied to stabilize the dilaton/axion
during inflation, and it is shown to lead to an instability. The instability is
cured by adding the alternative Fayet-Iliopoulos (FI) term that does not lead
to the gauged $R$-symmetry. Other stabilization mechanisms, based on the
Wess-Zumino (WZ)-type superpotential, are also studied in the presence of the
FI term. A possible connection to a D3-brane is briefly discussed too.
