The objective of the Arbitrary Lagrangian-Eulerian (ALE) methodology for solving multidimensional fluid flow problems is to move the computational mesh, using the flow as a guide, to improve the robustness, accuracy and efficiency of a simulation. The main elements in the ALE simulation are an explicit Lagrangian phase, a rezone phase in which a new mesh is defined, and a remapping (conservative interpolation) phase in which the Lagrangian solution is transferred to the new mesh. In most ALE codes, the main goal of the rezone phase is to maintain high quality of the rezoned mesh. In this paper, we describe a new rezone strategy which minimizes the $L_2$ norm of the solution error and maintains smoothness of the mesh. The efficiency of the new method is demonstrated with numerical experiments.