A bivariate Bayes method is proposed for estimating the mortality rates of a single disease for a given population, using additional information from a second disease. The information on the two diseases is assumed to be from the same population groups or areas. The joint frequencies of deaths for the two diseases for given populations are assumed to have a bivariate Poisson distribution with joint means proportional to the population sizes. The relationship between the mortality rates of the two different diseases if formulated through the twofold conditional autoregressive (CAR) model, where spatial effects as well as indexes of spatial dependence are introduced to capture the structured clusterings among areas. This procedure is compared to a univariate hierarchical Bayes procedure that uses information from one disease only. Comparisons of two procedures are made by the optimal property, a Monte Carlo study, real data, and the Bayes factor. All of the methods that we consider demonstrate a substantial improvement in the bivariate over the univariate procedure. For analyzing male and female lung cancer data from the state of Missouri, Markov chain Monte Carlo methods are used to estimate mortality rates.