In this method, the LNG stream is cooled using a refrigerating fluid in a first heat-exchanger. The refrigerating fluid is subjected to a second semi-open refrigeration cycle which is independent of the first cycle. The method comprises a step for introducing the stream of sub-cooled LNG into a distillation column and a step for recovering a stream of gas at the top of the column.The second refrigeration cycle comprises a step for forming a stream of refrigerating fluid from a portion of the top stream of gas, a step for compressing the stream of refrigerating fluid to a high pressure, then a step for expanding a portion of the stream of compressed refrigerating fluid in order to form a substantially liquid sub-cooling stream. The substantially liquid stream is evaporated in the first heat-exchanger.