Computational results of a two-dimensional gravity current in a lock-exchange flow is presented in which a finite volume of homogeneous fluid was released instantaneously into another fluid of slightly lower density when a lock gate was opened. The computations were performed in a two-dimensional channel by solving the incompressible Navier-Stokes equation for an inhomogeneous fluid, the continuity equation and the transport equation for solute by the finite volume method. For accurate representation of small density difference, the density variation relative to the characteristic density difference was used as one of primitive variables. The finite volume formulation holds the conservative property with respect to mass at the boundaries as well as at the density interface so that the total mass of the two fluids with different density remains constant. Some of standard numerical schemes were used to examine their performance to the density jump of the interface. The computed gravity current moves steadily in an initial phase, and the front speed decreases with distance in a self-similar phase when an internal bore on the interface reflected from the back of the lock reaches the front of the current. The effects of the bottom boundary layer on the internal structure of the current is investigated from numerical experiments with no slip and free-slip boundary conditions. The volume of the diluted fluid in a gravity current by the entrainment of ambient fluid is evaluated as a function of time to quantify the mixing. The result indicates that at low Reynolds numbers the subsequent mixing occurs in the early stage of the evolution of the gravity current in a lock-exchange flow.