This method includes introducing a downstream stream (140) of cracked gas from a downstream heat exchanger (58) in a downstream separator (60) and recovering, at the head of the downstream separator (60), a high-pressure fuel gas stream (144).The method includes the passage of the stream (144) of fuel through the downstream exchanger (58) and an intermediate exchanger (50, 54) to form a reheated high-pressure fuel stream (146), the expansion of the reheated high-pressure fuel stream (146) in at least a first dynamic expander (68) and the passage of the partially expanded fuel stream (148) from the intermediate exchanger (50, 54) in a second dynamic expander (70) to form an expanded fuel stream (152).The expanded fuel stream (152) from the second dynamic expander (70) is reheated in the downstream heat exchanger (58) and in the intermediate heat exchanger (50, 54).