The complex formation between a monolayer consisting of strong polyelectrolyte molecules covalently attached to a solid surface and an oppositely charged strong polyelectrolyte in solution is studied. The surface-attached polyelectrolyte molecules are grown in situ on the substrate using a surface-initiated polymerization ("grafting from") technique. The amount of oppositely charged polymer adsorbed by the polyelectrolyte monolayer on the surface ("brush") is measured as a function of salt concentration, deposition time, polymer concentration, and thickness of the surface-attached monolayer. When the oppositely charged polymer is added to the polyelectrolyte brush, which is swollen in salt-free water, the monolayer collapses rapidly due to formation of a nonstoichiometric, insoluble polyelectrolyte-polyelectrolyte complex. It is shown that polyelectrolyte brushes are an interesting substrate for the buildup of well-defined polyelectrolyte multilayers through a layer-by-layer deposition process.