  We calculate the bispectrum of primordial curvature perturbations, \zeta,
generated during "open inflation." Inflation occurs inside a bubble nucleated
via quantum tunneling from the background false vacuum state. Our universe
lives inside the bubble, which can be described as a
Friedman-Lema\^itre-Robertson-Walker (FLRW) universe with negative spatial
curvature, undergoing slow-roll inflation. We pay special attention to the
issue of an initial state for quantum fluctuations. A "vacuum state" defined by
a positive-frequency mode in de Sitter space charted by open coordinates is
different from the Euclidean vacuum (which is equivalent to the so-called
"Bunch-Davies vacuum"). Quantum tunneling then modifies the initial state away
from the original Euclidean vacuum. While most of the previous study on
modifications of the initial quantum state introduces, by hand, an initial time
at which the quantum state is modified as well as the form of the modification,
an effective initial time naturally emerges and the form is fixed by quantum
tunneling in open inflation models. Therefore, open inflation enables a
self-consistent computation of the effect of a modified initial state on the
bispectrum. We find a term which goes as <\zeta_{k_1} \zeta_{k_2} \zeta_{k_3}>
\propto 1/k_1^2k_3^4 in the so-called squeezed configurations, k_3\ll
k_1\approx k_2, in agreement with the previous study. The bispectrum in the
exact folded limit, e.g., k_1=k_2+k_3, is also enhanced and remains finite.
However, these terms are exponentially suppressed when the wavelength of \zeta
is smaller than the curvature radius of the universe. The leading-order
bispectrum is equal to the usual one from single-field slow-roll inflation; the
terms specific for open inflation arise only in the sub-leading order when the
wavelength of \zeta is smaller than the curvature radius.
