The gravitational effects associated with the radiative tail produced by a gravitational collapse with rotation are investigated. It is shown that the infinite blueshift of the tail's energy density occurring at the Cauchy horizon of the resulting black hole causes a classically unbounded inflation of the effective internal gravitational-mass parameter of the hole. Since this effect is causally disconnected from any external observer, the black-hole external mass remains bounded. The mass inflation phenomenon causes the spacetime curvature to grow to Planckian scales on a spacelike hypersurface in the vicinity of the Cauchy horizon, beyond which the classical laws of general relativity break down. A consequence is that an observer's trip to this hypersurface embraces all but the last Planck time of the entire black-hole classical history.