An apparatus (1) for sterilizing container closures comprises a conduit (2) defining a passage for the closure and having a wall (3) surrounding the passage, spraying means (4) for injecting inside the conduit (2) a sterilization fluid heated and vaporized, drying means for introducing hot air in the conduit (2) and means for heating the wall (3) of the conduit (2) operatively coupled thereto. A method for sterilizing container closures transported inside a conduit (2) having a wall (3) defining internally a passage for the closures comprises the steps of injecting inside the conduit (2) a sterilization fluid heated and vaporized, introducing hot air in the conduit (2) for drying the closures and heating the wall (3) of the conduit (2) by transferring heat directly to the conduit wall (3).