Combined facility for cryogenic separation and liquefaction of methane and carbon dioxide in a biogas stream, comprising: - a means M1 for mixing the biogas with a recycle gas R, - a compressor for compressing the mixture to the distillation pressure, - an exchanger E01 for cooling the compressed mixture, - a distillation column K01 supplied with the cooled mixture and making it possible to produce methane at the top of the column and a CO 2 -enriched liquid at the bottom of the column, - an exchanger E02 for liquefying the methane produced at the top of the column, - a means M2 for separating the liquefied methane into two portions: a "reflux" portion and a "product" portion, - a means M3 for expanding and heating the CO 2 -enriched liquid recovered at the bottom of the column and for recovering the cold from the CO 2 -enriched liquid, and - a separator vessel V01 for receiving the CO 2 -enriched stream from the means M3 and for recovering an overhead vapour and liquid CO 2, - a means for withdrawing vapour V1 from an intermediate stage of the distillation column K01, - a means for partially condensing the vapour V1 and producing a two-phase stream D1, - a means for reinjecting the two-phase stream D1 into the distillation column K01 at the stage corresponding to the equilibrium temperature, with - the means M1 such that the recycle gas R corresponds to the overhead vapour recovered at the outlet of the separator vessel V01, - the exchanger E01 and the means M3 being combined, and - the cryogenic separation unit comprising: • a distillation column K01 comprising a cold section at the top of the column and a hot section at the bottom of the column, • a means for physically separating the cold section and the hot section, • at least two containers V04 A/B external to the distillation column, for placing in contact the liquid from the cold section and the vapour rising from the hot section and for trapping all the solid CO 2, and • a device for regenerating the external containers, comprising a means for extracting a fluid from the distillation column K01 which is capable of liquefying the solid CO 2, a means for introducing this fluid into the external container(s) in regeneration so as to bring about the melting of the solid CO 2 and a means for reintroducing the resulting liquid CO 2 -vapour mixture into the distillation column K01. - a means M1 for mixing the biogas with a recycle gas R, - a compressor for compressing the mixture to the distillation pressure, - an exchanger E01 for cooling the compressed mixture, - a distillation column K01 supplied with the cooled mixture and making it possible to produce methane at the top of the column and a CO 2 -enriched liquid at the bottom of the column, - an exchanger E02 for liquefying the methane produced at the top of the column, - a means M2 for separating the liquefied methane into two portions: a "reflux" portion and a "product" portion, - a means M3 for expanding and heating the CO 2 -enriched liquid recovered at the bottom of the column and for recovering the cold from the CO 2 -enriched liquid, and - a separator vessel V01 for receiving the CO 2 -enriched stream from the means M3 and for recovering an overhead vapour and liquid CO 2, - a means for withdrawing vapour V1 from an intermediate stage of the distillation column K01, - a means for partially condensing the vapour V1 and producing a two-phase stream D1, - a means for reinjecting the two-phase stream D1 into the distillation column K01 at the stage corresponding to the equilibrium temperature, - the means M1 such that the recycle gas R corresponds to the overhead vapour recovered at the outlet of the separator vessel V01, - the exchanger E01 and the means M3 being combined, and - the cryogenic separation unit comprising: • a distillation column K01 comprising a cold section at the top of the column and a hot section at the bottom of the column, • a means for physically separating the cold section and the hot section, • at least two containers V04 A/B external to the distillation column, for placing in contact the liquid from the cold section and the vapour rising from the hot section and for trapping all the solid CO 2, and • a device for regenerating the external containers, comprising a means for extracting a fluid from the distillation column K01 which is capable of liquefying the solid CO 2, a means for introducing this fluid into the external container(s) in regeneration so as to bring about the melting of the solid CO 2 and a means for reintroducing the resulting liquid CO 2 -vapour mixture into the distillation column K01. • a distillation column K01 comprising a cold section at the top of the column and a hot section at the bottom of the column, • a means for physically separating the cold section and the hot section, • at least two containers V04 A/B external to the distillation column, for placing in contact the liquid from the cold section and the vapour rising from the hot section and for trapping all the solid CO 2, and • a device for regenerating the external containers, comprising a means for extracting a fluid from the distillation column K01 which is capable of liquefying the solid CO 2, a means for introducing this fluid into the external container(s) in regeneration so as to bring about the melting of the solid CO 2 and a means for reintroducing the resulting liquid CO 2 -vapour mixture into the distillation column K01.