This article explores a Bayesian analysis of a generalization of the Poisson distribution. By choice of a second parameter v, both under-dispersed and over-dispersed data can be modeled. The Conway-Maxwell-Poisson distribution forms an exponential family of distributions, so it has sufficient statistics of fixed dimension as the sample size varies, and a conjugate family of prior distributions. The article displays and proves a necessary and sufficient condition on the hyperparameters of the conjugate family for the prior to be proper, and it discusses methods of sampling from the conjugate distribution. An elicitation program to find the hyperparameters from the predictive distribution is also discussed.