This paper presents a novel method of nonlinear Kalman filtering, which unifies the best features of the accurate continuous-discrete extended and cubature Kalman filters. More precisely, the time updates in the discussed state estimator are done as those in the first filter whereas the measurement updates are conducted with use of the third-degree spherical-radial cubature rule applied for approximating the arisen Gaussian-weighted integrals. All this allows accurate predictions of the state mean and covariance matrix to be combined with accurate measurement updates. Moreover, the new filter is particularly effective for continuous-discrete stochastic systems with nonlinear and/or nondifferentiable observations. The efficiency of this mixed-type method is shown in comparison to the performance of the original accurate continuous-discrete extended and cubature Kalman filters in severe conditions of tackling a seven-dimensional radar tracking problem, where an aircraft executes a coordinated turn, with sufficiently long sampling times.