The present invention proposes hydrate process and apparatus for separating gas mixture continuously. The countercurrent continuous separation process in three serially connected stages is adopted to raise the CO2 recovering rate from fume, the CO2 concentration in the product and the separation rate and efficiency. Each of the separating systems consists of a hydrate synthesizer, a hydrate decomposer, a circulating pump, a heat exchanger, and a micro bubble mixing and jetting reactor. The present invention is suitable for continuous separation and enrichment of CO2 in the fume of coal burning power plant, and may be also used in the continuous separation and enrichment of gas mixture containing CO2 in chemical industry, metallurgy, petroleum and other fields.