This paper is aimed at stabilization of a cascade system composed of two subsystems. The first subsystem (ODE subsystem) is represented by a system of linear ordinary differential equations with a strictly upper triangular structure. The other subsystem (PDE subsystem) is described by a one-dimensional parabolic partial differential equation. The PDE subsystem is independent from the ODE subsystem, whereas the ODE subsystem depends on the PDE subsystem. The control input may affect both the subsystems. We develop a recursive procedure for designing stabilizing control laws based on the idea of the forwarding approach. The stability of the closed-loop system under the resulting control law is theoretically proved and numerically demonstrated.