The invention addresses the problem of edge bulging and breakout of as cast strip due to a decrease of perfect solid phase shell thickness of the strip caused by heat recuperation of the unsolidified portion of the strip, by providing a chamber 19 which encloses the travel path of a strip 8 delivered from a pair of chilled rolls 1a and 1b, a pair of chilled blocks 29a arranged in the chamber to be positioned near a nip between the chilled rolls 1a and 1b, with each chilled block 29a being shaped for loose fitting over an edge of a strip 8 and being capable of injecting cooling gas toward the edge of the strip 8. When the strip 8 is to be continuously cast, the interior of the chamber 19 provides an inert gas atmosphere, in which the cooling gas injected by the cooling blocks 29a injection-cools the edges of the strip 8 delivered from the chilled rolls 1a and 1b at a position near the nip between the chilled rolls 1a and 1b.