Finely divided, amphoteric, aqueous polymer dispersions which are obtainable by a two-stage polymerization, a prepolymer being prepared in the first polymerization stage from a monomer mixture comprising (a) from 15 to 40% by weight of at least one (meth)acrylate and/or (meth)acrylamide having in each case a free amino group, protonated amino group and/or quaternized amino group, (b) from 40 to 70% by weight of at least one optionally substituted styrene, (c) from 0 to 25% by weight of acrylonitrile and/or methacrylonitrile, (d) from 0.5 to 5% by weight of at least one ethylenically unsaturated monomer comprising acid groups and (e) from 0 to 30% by weight of at least one nonionic, ethylenically unsaturated monomer differing from (b) and (c), the sum of (a)+(b)+(c)+(d)+(e) being 100% by weight, in a solution polymerization in a solvent partly to completely miscible with water, water then being added to the solution/dispersion of the prepolymer and then at least one nonionic, ethylenically unsaturated monomer being subjected to an emulsion polymerization in the aqueous solution of the prepolymer in a second polymerization stage, processes for the preparation of the polymer dispersions by two-stage polymerization and use of the polymer dispersions as sizes for paper, board and cardboard.