In this paper, we propose an absolutely continuous bivariate probability distribution starting from two conditional distributions, which are univariate generalized exponential distributions. The resulting bivariate distribution has four parameters, which provide flexibility in analyzing various types of bivariate data sets. Several structural properties of the proposed bivariate distribution with generalized exponential conditionals are discussed. We derive the maximum likelihood estimators of the model parameters to facilitate the model fitting for bivariate data. Since the maximum likelihood estimators are not available in closed form, we explore the maximization of the log-likelihood function by using a four-dimensional optimization algorithm and a profile likelihood method. A Monte Carlo simulation study is used to evaluate the performance of the proposed parameter estimation procedures. To illustrate the applicability of the proposed bivariate distribution and the estimation methods, a real data set is analyzed.