We study a kind of delayed reaction-diffusion equation with Dirichlet boundary condition. We show the existence of a sequence of values {τ_kn}_n=0,1,2,... of the parameter τ such that a Hopf bifurcation occurs when the delay passes through each value {τ_kn}. The main techniques used here are some results on nonlinear eigenvalue problems, the analysis of the characteristic equation of the linearized problem, the Liapunov-Schmidt method and the implicit function theorem.