  We study the stability of a spherically symmetric perturbation around the
flat Friedmann-Lema$\hat{\i}$tre-Robertson-Walker spacetime in the ghost-free
bigravity theory, retaining nonlinearities of the helicity-$0$ mode of the
massive graviton. It has been known that, when the graviton mass is smaller
than the Hubble parameter, homogeneous and isotropic spacetimes suffer from the
Higuchi-type ghost or the gradient instability against the linear perturbation
in the bigravity. Hence, the bigravity theory has no healthy massless limit for
cosmological solutions at linear level. In this paper we show that the
instabilities can be resolved by taking into account nonlinear effects of the
scalar graviton mode for an appropriate parameter space of coupling constants.
The growth history in the bigravity can be restored to the result in general
relativity in the early stage of the Universe, in which the St\"uckelberg
fields are nonlinear and there is neither ghost nor gradient instability.
Therefore, the bigravity theory has the healthy massless limit, and cosmology
based on it is viable even when the graviton mass is smaller than the Hubble
parameter.
