We present a simple model for the binding of persistence-length segments of a polyelectrolyte to an oppositely charged spherical macroion with radius of the same order as the polymer persistence length. The polyelectrolyte is taken as a rod with elastic bending capacity. The macroion is abstracted as an absolutely rigid circular arc. In the bound state, the polyelectrolyte bends onto the macroion, and some of the opposite charges are mutually neutralized. The curvature of the macroion imposes stability limits on the bound state. The model predicts transitions to bound-state stability in the same conditions (polyelectrolyte charge density and bare persistence length, macroion radius, salt concentration) under which they are observed in experimental studies of the binding of polyelectrolytes to oppositely charged micelles. The calculations also suggest why macroscopic phase separation occurs in experiments below a critical salt concentration but not above.