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It is well known that it is possible to represent a Birnbaum-Saunders variable as a relatively simple (and invertible) function of a standard normal random variable. Marginal transformations of this kind are applied in this paper to a bivariate distribution with generalized skew-normal conditionals (and normal marginals), to obtain a new bivariate Birnbaum-Saunders distribution. Parameter estimation for this model is implemented using an EM algorithm. A simulation study sheds light on the performance of the estimation strategy. Data from a cancer risk study is used to illustrate use of the model. For this data set, the new model exhibits better performance than does a competing skew-normal based model already discussed in the literature. Possible multivariate extensions of the new model are outlined.