This dissertion presents a new technique for modifying all balanced incomplete block designs (BIBD) to provide orthogonal estimates when the BIBD is to be used as a weighing design. Previously, K.S. Banerjee developed a method for modifying BIBD to provide orthogonal estimates. However, for a certain class of BIBD, Banerjee's method fails to provide orthogonal estimates. A comparison of the relative efficiencies of the new procedure with that of Banerjee's procedure is also presented. In addition, under the new procedure it is shown that the covariance matrix of the estimators obtained by the least squares procedure is identical to that obtained by the maximum likelihood procedure, even when the design matrix X is not square. Several examples of the utilization of the new technique, along with a historical development of the weighing problem from its origin in a casual example by Yates through the work of Hotelling, Mood, Kempthorne, and Banerjee as relative to the problem of providing orthogonal estimates, are also presented.