A method and an apparatus for transporting a rapidly quenched ribbon from a cooling roll to a winding reel by a pinch roll are disclosed. The pinch roll comprises a brush roll and a solid roll for transporting the ribbon such as to maintain an adequate tension in the ribbon without any transversal displacement and breakdown of ribbon. The tension is controlled within a range of 2.about.8 kgf by adjusting at least one of an amount of pressing of the brush roll against the solid roll, a pressing force on the brush roll and a ratio between circumferential speeds of the pinch roll and the cooling roll.