The problem addressed here is concerned with estimating individual weights of objects by using a chemical balance weighing design under the constraint on the number in which each object is weighed. The model for a chemical balance design is \(y(n\times 1)= x(n\times p)w(p\times 1)+ e(n\times 1)\), when \(y\) is a vector of observations, \(x\) is the known design matrix with elements \(-1\), \(0\), \(1\) \((p\leq n)\), \(w\) is a column vector of unknown parameters \(w_j\) \((j= 1,2,3,\dots, p)\), and \(e\) is a vector of unobserved random errors with \(E(e)= o(n\times 1)\), and \(\text{Var}(e)= \overset {2}\sigma I(n\times n)\). The normal equations for estimating \(w\) are \(x'x\widehat w= x'y\). A chemical balance weighing design is singular (nonsingular) if the corresponding matrix \(x'x\) is singular (nonsingular). It is well known that the matrix \(x'x\) is nonsingular if and only if the matrix \(x\) is of full column rank \((=p)\). If \(x'x\) is nonsingular, then \(\widehat w= (x'x)^{-1}x'y\), and the variance-covariance matrix of the estimates is \(\text{Var}(\widehat w)= \overset {2}\sigma(x'x)^{-1}\). The authors here address the problem of constructing the design matrix \(x\) for an optimum chemical balance weighing design under the constraint on the number in which each object is weighed. A result on a lower bound for the variance of each of the estimated weights resulting from the use of the chemical balance weighing design is given and a necessary and sufficient condition for obtaining this lower obund is also presented. Ternary balanced block designs with parameters \(v\), \(b\), \(r\), \(k\), \(\lambda\), \(\rho_1\), \(\rho_2\) are used in constructing the optimum chemical balance weighing designs. Finally the authors present the list of all parameter combinations of the ternary balanced block designs with \(r\leq 10\) which give optimum chemical balance weighing designs.