A process for preparing butadiene from n-butenes, comprising the steps of: A) providing an input gas stream a comprising n-butenes, B) feeding the input gas stream a comprising n-butenes and at least one gas containing least oxygen into at least one oxidative dehydrogenation zone and oxidatively dehydrogenating n-butenes to butadiene, giving a product gas stream b comprising butadiene, unconverted n-butenes, water vapor, oxygen, low-boiling hydrocarbons and high-boiling secondary components, with or without carbon oxides and with or without inert gases; Ca) cooling the product gas stream b by contacting with a cooling medium in at least one cooling zone, the cooling medium being at least partly recycled and having an aqueous phase and an organic phase, Cb) compressing the cooled product gas stream b which may have been depleted of high-boiling secondary components in at least one compression stage, giving at least one aqueous condensate stream c1 and one gas stream c2 comprising butadiene, n-butenes, water vapor, oxygen and low-boiling hydrocarbons, with or without carbon oxides and with or without inert gases; D) removing uncondensable and low-boiling gas constituents comprising oxygen and low-boiling hydrocarbons, with or without carbon oxides and with or without inert gases, as gas stream d2 from the gas stream c2 by absorbing the C 4 hydrocarbons comprising butadiene and n-butenes in an absorbent, giving an absorbent stream laden with C 4 hydrocarbons and the gas stream d2, and then desorbing the C 4 hydrocarbons from the laden absorbent stream, giving a C 4 product gas stream d1, E) separating the C 4 product stream d1 by extractive distillation with a butadiene-selective solvent into a stream e1 comprising butadiene and the selective solvent and a stream e2 comprising n-butenes; F) distilling the stream e1 comprising butadiene and the selective solvent into a stream f1 consisting essentially of the selective solvent and a stream f2 comprising butadiene, wherein stage Cb) comprises at least two compression stages Cba) and at least two cooling stages Cbb) configured in the form of quench columns, the cooling in the cooling stages being effected by direct contacting with a biphasic cooling medium having an aqueous phase and an organic phase from an organic solvent, the organic solvent being selected from the group consisting of toluene, o-, m- and p-xylene, mesitylene, mono-, di- and triethylbenzene, mono-, di- and triisopropylbenzene and mixtures thereof and, in the cooling stages Cbb), the mass ratio of the aqueous phase to the organic phase in the cooling medium when it is fed into the cooling zones prior to the contacting with the product gas stream being from 0.15:1 to 10:1. A) providing an input gas stream a comprising n-butenes, B) feeding the input gas stream a comprising n-butenes and at least one gas containing least oxygen into at least one oxidative dehydrogenation zone and oxidatively dehydrogenating n-butenes to butadiene, giving a product gas stream b comprising butadiene, unconverted n-butenes, water vapor, oxygen, low-boiling hydrocarbons and high-boiling secondary components, with or without carbon oxides and with or without inert gases; Ca) cooling the product gas stream b by contacting with a cooling medium in at least one cooling zone, the cooling medium being at least partly recycled and having an aqueous phase and an organic phase, Cb) compressing the cooled product gas stream b which may have been depleted of high-boiling secondary components in at least one compression stage, giving at least one aqueous condensate stream c1 and one gas stream c2 comprising butadiene, n-butenes, water vapor, oxygen and low-boiling hydrocarbons, with or without carbon oxides and with or without inert gases; D) removing uncondensable and low-boiling gas constituents comprising oxygen and low-boiling hydrocarbons, with or without carbon oxides and with or without inert gases, as gas stream d2 from the gas stream c2 by absorbing the C 4 hydrocarbons comprising butadiene and n-butenes in an absorbent, giving an absorbent stream laden with C 4 hydrocarbons and the gas stream d2, and then desorbing the C 4 hydrocarbons from the laden absorbent stream, giving a C 4 product gas stream d1, E) separating the C 4 product stream d1 by extractive distillation with a butadiene-selective solvent into a stream e1 comprising butadiene and the selective solvent and a stream e2 comprising n-butenes; F) distilling the stream e1 comprising butadiene and the selective solvent into a stream f1 consisting essentially of the selective solvent and a stream f2 comprising butadiene,