We present a theory that describes equilibrium morphologies of micelles formed by diblock copolymer with one neutral and one weakly dissociating polyelectrolyte block in dilute aqueous solution. We demonstrate that, depending on the copolymer composition, variations in the pH and the ionic strength in the solution can trigger different sequences of morphological transitions. At high solution salinity, an increase in ionic strength gives rise to sphere-to-cylinder and cylinder-to-lamella transitions. At low solution salinity, the sequence of morphological transformations is reversed, and micelles undergo lamella-to-cylinder and cylinder-to-sphere transitions upon an increase in the salt concentration. We obtain the diagrams of states of diblock copolymer solution and derive analytical expressions for binodal lines corresponding to the first-order phase transitions and the coexistence of aggregates of different morphologies. Our results provide the guidelines to create block polyelectrolytes aggregates with stimuli-induced polymorphism.