The ergodicity problem of the Wonham filter as a diffusion process is discussed. Since the Wonham equation is degenerate, we apply an ergodic theorem of degenerate Markov diffusions to the problem. Under a certain condition, the Wonham equation satisfies Hormander's condition, and the Wonham filter has a continuous transition density. From these results, we obtain that the Wonham filter is uniformly ergodic as a diffusion process.