A finned stainless steel alloy foil pipe heat exchanger and a making method thereof are provided. The heat exchanger is formed by radiating fins (2), which are made of alloy or nonferrous metal and are separately and closely arranged, and a stainless steel alloy foil coiled pipe (1) whose wall thickness is 0.1mm~0.3mm. The making method thereof includes preparing material, making rims (14) of pipe ends of bent pipes (12) or straight pipes (11), fixing the fins (2) on the straight pipes (11), and welding.