  We analytically study the non-linear stability of a spherically symmetric
wormhole supported by an infinitesimally thin brane of negative tension, which
has been devised by Barcelo and Visser. We consider a situation in which a thin
spherical shell composed of dust falls into an initially static wormhole; The
dust shell plays a role of the non-linear disturbance. The self-gravity of the
falling dust shell is completely taken into account through Israel's formalism
of the metric junction. When the dust shell goes through the wormhole, it
necessarily collides with the brane supporting the wormhole. We assume the
interaction between these shells is only gravity and show the condition under
which the wormhole stably persists after the dust shell goes through it.
