The typical elements in a numerical simulation of fluid flow using moving meshes are a time integration scheme, a rezone method in which a new mesh is defined, and a remapping (conservative interpolation) in which a solution is transferred to the new mesh. The objective of the rezone method is to move the computational mesh to improve the robustness, accuracy and eventually efficiency of the simulation. In this paper, we consider the one-dimensional viscous Burgers' equation and describe a new rezone strategy which minimizes the $L_2$ norm of error and maintains mesh smoothness. The efficiency of the proposed method is demonstrated with numerical examples.