We present designs for exponential stabilization of an ordinary differential equation (ODE)-heat partial differential equation (PDE)-ODE coupled system where the control actuation only acts in one ODE. The combination of PDE backstepping and ODE backstepping is employed in a state feedback control law and in an observer that estimates PDE and two ODE states using only one PDE boundary measurement. Based on the state feedback control law and the observer, the output feedback control law is then proposed. The exponential stability of the closed-loop system and the boundedness and exponential convergence of the control law are proved via Lyapunov analysis. Finally, numerical simulations validate the effectiveness of this method for the "sandwiched" system.