For a recurrent neural network (RNN), its transient response is a critical issue, especially for real-time signal processing applications. The conventional RNN training algorithms, such as backpropagation through time (BPTT) and real-time recurrent learning (RTRL), have not adequately addressed this problem because they suffer from low convergence speed. While increasing the learning rate may help to improve the performance of the RNN, it can result in unstable training in terms of weight divergence. Therefore, an optimal tradeoff between RNN training speed and weight convergence is desired. In this paper, a robust adaptive gradient-descent (RAGD) training algorithm of RNN is developed based on a novel RNN hybrid training concept. It switches the training patterns between standard real-time online backpropagation (BP) and RTRL according to the derived convergence and stability conditions. The weight convergence and L2-stability of the algorithm are derived via the conic sector theorem. The optimized adaptive learning maximizes the training speed of the RNN for each weight update without violating the stability and convergence criteria. Computer simulations are carried out to demonstrate the applicability of the theoretical results.