In deep oil drilling, the length of the domain over which the wave equation models the torsional dynamics of the drill string keeps changing with time, and it also depends on the drill bit speed. Moreover, the drill bit speed cannot be controlled directly. In this context, we consider predictor-based design for the cascade system of a nonlinear ODE and a wave PDE with a moving uncontrolled boundary. In comparison with prior results on wave PDE-ODE cascades, this work differs by giving rise to a prediction horizon that is not given explicitly but has to be found from an implicit relationship involving the delay function and the future solution of the system. Stability analysis of the closed-loop system is conducted by constructing infinite-dimensional backstepping transformations and a Lyapunov functional. An explicit feedback law for compensating the wave actuator dynamics is obtained. For the moving boundary that depends on both the ODE's state and time, a region of attraction is estimated. For the moving boundary that depends on time, a global stabilization for the closed-loop system is achieved. Finally, an example is given to illustrate the effectiveness of the proposed design technique. (C) 2016 Elsevier Ltd. All rights reserved.