A simple model system was used to examine the complexation between a charged micelle and an oppositely charged polyelectrolyte. Structural data of the micelle-polyelectrolyte complex and thermodynamic quantities of the complexation as a function of the polyelectrolyte linear charge density at two different chain flexibilities were calculated from Monte Carlo simulations and thermodynamic integration. The presence of the polyelectrolyte was found to reduce the critical micellar concentration, i.e., the lowest surfactant concentration at which micelles are formed. The largest reduction of the cmc occurred for the highest linear charge density and the most flexible polyelectrolyte (110 times smaller), whereas the smallest reduction occurred for the lowest linear charge density and the more rigid polyelectrolyte (10 times). The complexation is caused by the strong attractive electrostatic interaction between the micelle and the polyelectrolyte and is manifested by (i) a decrease of the electrostatic energy due to small micelle-polyelectrolyte charge separations and (ii) an increase of the entropy due to the release of the counterions of the macroions. The predicted results are in accordance with experimental data of systems with different linear charge densities and chain flexibilities of the polyelectrolyte.