Disclosed is a process for the purification of an organic composition (OC1) by adsorption using an assembly containing at least two adsorbers. The organic composition (OC1) comprising at least one alkane, at least one olefin and at least one compound containing oxygen and/or sulphur is fed into a first adsorber (A1) of the assembly in order to obtain an organic composition (OC2) comprising at least one alkane, at least one olefin and a reduced amount of at least one compound containing oxygen and/or sulphur compared to the respective amount in organic composition (OC1). Hydrogenation of the organic composition (OC2) provides a stream (S2) comprising at least one alkane and a reduced amount of at least one olefin compared to the respective amount in organic composition (OC2) obtained after feeding into the first adsorber (A1). A second adsorber (A2) of the assembly is regenerated by contact with stream (S2).