Continuous method for the production of polyethylene furanoate or polybutylene furanoate having an average molecular weight, determined by means of GPC, for the weight average Mw of at least 30,000 Da, in which a) at least one heteroaromatic dicarboxylic acid selected from the group consisting of 2,5-furandicarboxylic acid, 3,4-furandicarboxylic acid, 2,4-furandicarboxylic acid, 2,3-furandicarboxylic acid and/or combinations or mixtures hereof or the diesters or acid anhydrides thereof and at least one alcohol selected from the group consisting of ethylene glycol, diethylene glycol, triethylene glycol, 1,3-butanediol, 1,4-butanediol and/or combinations or mixtures hereof are processed by mixing to form a paste or by mixing at temperatures of 10°C to 250°C to form a homogeneous solution, b) the paste or solution obtained in step a) is converted to form an esterification product which comprises at least one diester and/or at least one oligoester and which is formed from the at least one heteroaromatic dicarboxylic acid with the at least one alcohol, and c) the reaction product obtained from step b) is polycondensed or copolycondensed at reduced pressure relative to normal conditions, characterised in that the poly- or copolycondensation step c) is implemented in two steps, there being produced, in a first step c 1) from the reaction product obtained from step b) by polycondensation or copolycondensation, a polyester prepolymer or copolyester prepolymer at a pressure of 0.1 bar to 2 bar, and there being obtained, in a subsequent step c 2), from the polyester prepolymer or copolyester prepolymer from step c 1) by polycondensation or copolycondensation at a pressure of 0.1 mbar to 30 mbar, the polyester or copolyester, the reaction product produced in step b) being adjusted to an intrinsic viscosity (I.V.) of 0.04 dl/g to 0.12 dl/g measured according to ISO 1628-5 and the intrinsic viscosity of the polyester after step c 2) being 0.60 dl/g to 2.0 dl/g. a) at least one heteroaromatic dicarboxylic acid selected from the group consisting of 2,5-furandicarboxylic acid, 3,4-furandicarboxylic acid, 2,4-furandicarboxylic acid, 2,3-furandicarboxylic acid and/or combinations or mixtures hereof or the diesters or acid anhydrides thereof and at least one alcohol selected from the group consisting of ethylene glycol, diethylene glycol, triethylene glycol, 1,3-butanediol, 1,4-butanediol and/or combinations or mixtures hereof are processed by mixing to form a paste or by mixing at temperatures of 10°C to 250°C to form a homogeneous solution, b) the paste or solution obtained in step a) is converted to form an esterification product which comprises at least one diester and/or at least one oligoester and which is formed from the at least one heteroaromatic dicarboxylic acid with the at least one alcohol, and c) the reaction product obtained from step b) is polycondensed or copolycondensed at reduced pressure relative to normal conditions, characterised in that the poly- or copolycondensation step c) is implemented in two steps, there being produced, in a first step c 1) from the reaction product obtained from step b) by polycondensation or copolycondensation, a polyester prepolymer or copolyester prepolymer at a pressure of 0.1 bar to 2 bar, and there being obtained, in a subsequent step c 2), from the polyester prepolymer or copolyester prepolymer from step c 1) by polycondensation or copolycondensation at a pressure of 0.1 mbar to 30 mbar, the polyester or copolyester, the reaction product produced in step b) being adjusted to an intrinsic viscosity (I.V.) of 0.04 dl/g to 0.12 dl/g measured according to ISO 1628-5 and the intrinsic viscosity of the polyester after step c 2) being 0.60 dl/g to 2.0 dl/g.