This paper addresses an accurate and effective implementation of the continuous-discrete extended Kalman filtering method. The technique under discussion is grounded in numerical solution of the moment differential equations to predict the state mean of the stochastic dynamical system and the corresponding error covariance matrix. Here, we apply an efficient embedded Runge-Kutta pair possessing superior stability and many other attractive features, including automatic global error control, in order to improve performance of the complex computational procedure consisting of the extended Kalman filter and the underlying adaptive ODE solver. Thus, we introduce a new continuous-discrete adaptive extended Kalman filter and show its advantage over the standard variant on two test examples. In practice, this technique allows for much longer sampling intervals without any loss of accuracy, and that improves the applied potential of the extended Kalman filtering method, significantly.