Abstract: Based on my recent writings[1～ 5] about the colouring of a planar graph, this paper first analyzes the causes for the mistakes of A. B. Kempe's proof of the Four-colour conjecture, then suggests a way to correct them, finally presents a new proof of the Four-colour conjecture, which is complete and exact.

Key words: planar graph, kempe 's colourinterchange method, transferring colourinterchange method, four-colour problem