In the preparation of a rubber-modified molding material, in a first stage, a first mixture (A) contains at least one alkyl acrylate or methacrylate (a) of the formula (I) ##STR1## where R.sup.1 is hydrogen or methyl and R.sup.2 is alkyl of 1 to 32 carbon atoms, if required a first monomer (b) having two or more double bonds and, if required, also a second monomer (c) or a plurality thereof is polymerized, preferably with free radicals, to give a rubber (B), in a second stage, a third monomer (d) or a plurality thereof is added to the resulting rubber (B) to form a second mixture (C), whereby the rubber (B) is dissolved or swollen in the third monomer (d), with or with the addition of a solvent, and in a third stage, the second mixture (C) formed is polymerized with free radicals to give the rubber-modified molding material (D), and the mixture is reacted at >120.degree. C., during or after this polymerization, with a compound (e) which contains at least two peroxy groups.