We propose an approach for simultaneous topology identification and synchronization of a complex dynamical network with directed interconnections that relies on the edge-agreement framework and on adaptive-control approaches by design of an auxiliary synchronizing network. Our method guarantees the identification of the unknown directed topology without the need for verifying the Linear Independence Conditions normally required by previous works in the literature. Furthermore, it also guarantees that the complex network reaches synchronization as determined by the internal dynamics of the system. Under our identification algorithm we provide strong stability results for the estimation errors in the form of uniform semiglobal practical asymptotic stability of the estimation errors. Finally, we demonstrate the effectiveness of our approach with a numerical example.