The invention concerns a copolymer of ethylene oxide and at least one substituted oxirane carrying a cross-linkable function. The copolymer comprises ethylene oxide, —O—CH2—CHR— units in which R is a substituent containing a reactive function which is cross-linkable by free radical process, R may be different from one unit to the other, and possibly —O—CH2—CHR′— units in which R′ is a substituent containing no reactive function which is cross-linkable by means of a free radical process, R′ may be different from one unit to the other. It is characterized by an excellent polymolecularity index I=Mw/Mn and a statistical distribution of the different monomer units. The copolymer is prepared by an anionic copolymerization process. The copolymer is useful for preparing a solid electrolyte having good mechanical properties, a good cationic conductivity and a good chemical compatibility with the electrodes of a generator operating with alkali metals such as lithium and sodium.