Among many bivariate exponential distributions proposed in the statistical literature, the BVE of Marshall and Olkin (1967) is widely accepted. An important property of the BVE is that if X and Y are BVE, then X = Y with positive probability, which makes the BVE inappropriate in many practical situations. For example, in certain types of diseases where the occurrence of simultaneous failure of a pair of organs is rare, an appropriate bivariate exponential distribution to describe the failure of these paired organs should be one that is absolutely continuous. The BVE, however, possesses some other properties that have useful physical interpretations. Therefore, the derivation of an absolutely continuous bivariate exponential distribution retaining the desirable properties of the BVE appears to be a worthwhile objective. To this end, we consider a characterization property of the BVE and modify it suitably to derive an absolutely continuous bivariate distribution. The important characterization property of the BVE random variable (X, Y), proved by Block (1977), is that the following results are true: (i) X and Y are marginally exponential, (ii) min(X, Y) is exponential, and (iii) min(X, Y) is independent of X – Y. This characterization property is modified in this article as follows. We require absolute continuity of the distribution of (X, Y) and replace condition (iii) by the condition of independence of min(X, Y) and some function g(X, Y), which, like X – Y in the BVE, is such that g(X, X) = 0, g(X, Y) is strictly increasing (decreasing) in X(Y) for fixed Y(X). It is also assumed that g(X, Y) has a specified distributional form. An absolutely continuous bivariate exponential distribution, called the ACBVE2, is derived here by using this modified characterization property. This characterization property will lead to the absolutely continuous bivariate distribution of Block and Basu (1974) if one works with marginals that are “weighted averages” of exponential distributions. The ACBVE2 has a density that is of product type on each of {x < y} and {x > y} regions of the first quadrant. Some distributional properties of the ACBVE2 are presented. These include a characterization through the independence of min(X, Y) and X – Y + k(X, Y) for some k(X, Y), and a representation in terms of independent random variables. The discrepancy between the ACBVE2 and the BVE is studied. Two sets of method-of-moments–type estimates are considered, and their performances are compared using a Monte Carlo simulation.