The possibility of localized inflation is investigated by calculating the dynamics of a spherically symmetric region of false vacuum which is separated by a domain wall from an infinite region of true vacuum. For a range of initial conditions, the false-vacuum region will undergo inflation. An observer in the exterior true-vacuum region will describe the system as a black hole, while an observer in the interior will describe a closed universe which completely disconnects from the original spacetime. We suggest that this mechanism is likely to lead to an instability of Minkowski space: a region of space might undergo a quantum fluctuation into the false-vacuum state, evolving into an isolated closed universe; the black hole which remains in the original space would disappear by quantum evaporation. The formation of these isolated closed universes may also be relevant to the question of information loss in black-hole formation.