A process for the preparation of polyalkylene glycols of the general formulae (Ia) or (Ib) or mixtures thereof: in which R 1 is C 1 -C 6 -alkyl, each R 2, independently of one another, is 1,2-alkanediyl having 2 to 4 carbon atoms; B is hydrogen or the group where A is bonded to the radical of the molecule; R 4 is hydrogen, linear or branched C 1 -C 10 -alkyl, C 2 -C 10 -alkenyl; R 3 and R 5, independently of one another, are linear or branched C 1 -C 10 -alkyl, C 2 -C 10 -alkenyl, where R 3 and R 5, also together with the carbon atoms to which they are bonded, can form a 4- to 8-membered saturated or unsaturated carbocycle which is substituted by where A is bonded to the radical of the molecule and where the carbocycle can optionally carry further substituents selected from C 1 -C 10 -alkyl and C 2 -C 10 -alkenyl groups; with the proviso that if B is hydrogen, R 3 and R 5, together with the carbon atoms to which they are bonded, form the 4- to 8-membered saturated or unsaturated carbocycle; A is a single bond or C 1 -C 10 -alkylene; R 6 is hydrogen, linear or branched C 1 -C 10 -alkyl, C 2 -C 10 -alkenyl; n is 1 to 100; comprising the following steps: a) provision of a compound R 1 -[O-R 2 -] n -OH, and b) reaction of R 1 -[O-R 2 -] n -OH from step a) in the presence of a catalyst with an epoxide or an epoxide equivalent of the formula to give compounds of the formulae (Ia) or (Ib) or mixtures thereof, where R 1, R 2, R 3, R 4, R 5 and B in the formulae and have the meanings given above and m is 0 or 1. R 1 is C 1 -C 6 -alkyl, each R 2, independently of one another, is 1,2-alkanediyl having 2 to 4 carbon atoms; B is hydrogen or the group where A is bonded to the radical of the molecule; R 4 is hydrogen, linear or branched C 1 -C 10 -alkyl, C 2 -C 10 -alkenyl; R 3 and R 5, independently of one another, are linear or branched C 1 -C 10 -alkyl, C 2 -C 10 -alkenyl, where R 3 and R 5, also together with the carbon atoms to which they are bonded, can form a 4- to 8-membered saturated or unsaturated carbocycle which is substituted by where A is bonded to the radical of the molecule and where the carbocycle can optionally carry further substituents selected from C 1 -C 10 -alkyl and C 2 -C 10 -alkenyl groups; with the proviso that if B is hydrogen, R 3 and R 5, together with the carbon atoms to which they are bonded, form the 4- to 8-membered saturated or unsaturated carbocycle; where R 3 and R 5, also together with the carbon atoms to which they are bonded, can form a 4- to 8-membered saturated or unsaturated carbocycle which is substituted by where A is bonded to the radical of the molecule and where the carbocycle can optionally carry further substituents selected from C 1 -C 10 -alkyl and C 2 -C 10 -alkenyl groups; with the proviso that if B is hydrogen, R 3 and R 5, together with the carbon atoms to which they are bonded, form the 4- to 8-membered saturated or unsaturated carbocycle; A is a single bond or C 1 -C 10 -alkylene; R 6 is hydrogen, linear or branched C 1 -C 10 -alkyl, C 2 -C 10 -alkenyl; n is 1 to 100; comprising the following steps: a) provision of a compound R 1 -[O-R 2 -] n -OH, and b) reaction of R 1 -[O-R 2 -] n -OH from step a) in the presence of a catalyst with an epoxide or an epoxide equivalent of the formula to give compounds of the formulae (Ia) or (Ib) or mixtures thereof, a) provision of a compound R 1 -[O-R 2 -] n -OH, and b) reaction of R 1 -[O-R 2 -] n -OH from step a) in the presence of a catalyst with an epoxide or an epoxide equivalent of the formula to give compounds of the formulae (Ia) or (Ib) or mixtures thereof, where R 1, R 2, R 3, R 4, R 5 and B in the formulae and have the meanings given above and m is 0 or 1.