In this note, a stabilizing feedback boundary controller for the equilibrium solutions to parabolic equations with Dirichlet boundary control is designed. The feedback controller is expressed in terms of the eigenfunctions phi(j) corresponding to unstable eigenvalues {lambda(j)}(j=1)(N) of the linearized equation. For d = 1, this stabilizing procedure is applicable for N = 1 only, while for d > 1 it is of conditional nature requiring the independence of partial derivative phi(j)/partial derivative n on the part of the boundary where the control is applied.