  We investigate the recently proposed natural double inflation model in
supergravity. Chaotic inflation first takes place by virtue of the
Nambu-Goldstone-like shift symmetry. During chaotic inflation, an initial value
of second inflation (new inflation) is set, which is adequately far from the
local maximum of the potential due to the small linear term in the K\"ahler
potential. Then, primordial fluctuations within the present horizon scale may
be produced during both inflations. Primordial fluctuations responsible for
anisotropies of the cosmic microwave background radiation and the large scale
structure are produced during chaotic inflation, while fluctuations on smaller
scales are produced during new inflation. Because of the peculiar nature of new
inflation, they can become as large as $10^{-1}$-$10^{-2}$, which may lead to
the formation of primordial black holes.
