This paper discusses the Proportional Integral (PI) regulation control of the left Neumann trace of a one-dimensional reaction-diffusion equation with a delayed right Dirichlet boundary control. Specifically, a PI controller is designed based on a finite-dimensional truncated model that captures the unstable dynamics of the original infinite-dimensional system. In this setting, the control input delay is handled by resorting to the Artstein transformation. The stability of the resulting infinite-dimensional system, as well as the tracking of a constant reference signal in the presence of a constant distributed perturbation, is assessed based on the introduction of an adequate Lyapunov function. The theoretical results are illustrated with numerical simulations.