Process for copolymerizing ethylene and esters of vinyl alcohol in the presence of free-radical polymerization initiators at pressures in the range of from 110 MPa to 500 MPa and temperatures in the range of from 100° C. to 350° C. in a continuously operated polymerization apparatus comprising a polymerization reactor and one or more compressors, which compress the monomer mixture fed to the polymerization reactor to the polymerization pressure, wherein the monomer mixture is compressed by a sequence of compression stages in which the compressed gas mixture is cooled after each compression stage and the fraction of the monomer mixture, which is liquid after this cooling, is separated off and returned to the polymerization apparatus in liquid form, and wherein at least a part of the liquid fractions obtained after compressing the monomer mixture in the respective compression stage to a pressure of from 0.2 MPa to 10 MPa is purified before being returned to the polymerization process.