We present a mean-field theory to describe micelles formed by diblock copolymer with one neutral and one polyelectrolyte block in dilute aqueous solution. We examine equilibrium parameters of micelles as a function of degree of polymerization of both blocks, fraction of charged monomers in the ionic block, and the ionic strength in the solution. In contrast to earlier scaling studies, our theory enables us to construct phase diagrams of the system and distinguish the ranges of thermodynamic stability of spherical, cylindrical, and lamellar aggregates. We demonstrate that spherical starlike or crew-cut micelles are thermodynamically stable in a range of moderate salt concentrations. The morphological sphere-to-cylinder and cylinder-to-lamella transitions of the aggregates are predicted to occur for crew-cut micelles upon an increase in salt concentration or/and a decrease in degree of ionization of corona (ionic) block. The driving force for morphological transitions is the gain in conformational entropy of nonionic core block.