This paper studies the problem of estimation of the total weight of objects using a chemical balance weighing design under the restriction |L−R| ≤a, whereL andR represent the number of objects placed on the left and right pans, respectively. A lower bound for the variance of the estimated total weight is given and a necessary and sufficient condition for this lower bound to be attained is obtained. Finally, weighing designs for which this lower bound is attainable are constructed.