High molecular weight polyamides are prepared by (a) polymerizing monomers selected from the group consisting of aminonitriles of the formula I and mixtures of diamines of the formula II with dinitriles of the formula III where m, n and p are integers from two to twelve, in a molar diamine/dinitrile ratio of from 0.98:1 to 1.02:1 to give a prepolymer capable of being granulated, (b) granulating the prepolymer obtained under (a) and (c) subjecting the granules obtained under (b) to solid-phase postcondensation at from 5.degree. to 100.degree. C. below the melting point of the prepolymer in the presence of an inert gas which contains at least 50% by volume of superheated steam to give a high molecular weight polyamide having a relative viscosity of from 2.3 to 5.0, the difference between the relative viscosity of the resulting prepolymer from stage (a) and that of the high molecular weight polyamide from stage (c) being at least 0.4.