A stability analysis of fronts is performed analytically for a Rinzel-Keller-type model with equal diffusion constants. Exact solutions are obtained for the propagating front, the bifurcation diagram of the front velocity and the growth rate of disturbances. The effect of the most unstable eigenmode on the front shape is displayed. Moreover, we give the eigenvalues of the stability operator and show explicitly how a non-moving front becomes unstable at the bifurcation point while a moving front becomes stable.