Unique-word (UW) OFDM is a new multicarrier technique that was recently proposed in [M. Huemer, C. Hofbauer, and J. B. Huber, “The Potential of Unique Words in OFDM, Proc. InOWo, Hamburg, Germany, Sep. 2010, pp. 140-144]. In this technique, the guard interval that contains a priori known samples is part of the FFT interval. To be able to construct the UW-OFDM signal, some carriers must be sacrificed to transmit redundant information-these carriers are called the redundant carriers. It turns out that the average redundant energy needed strongly depends on the positions of the redundant carriers. In this paper, we look for the redundant carrier placement that minimizes the average redundant energy. However, this optimization is a NP-hard problem. Therefore, we reformulate the optimization problem and implement a branch and bound (B&B) algorithm to find the optimum. The results of the B&B search algorithm are compared to a theoretical lower bound, and with the results for two low cost redundant carrier placements, i.e., the quasi-uniform (QU) distribution [H. Steendam, “The Quasi-Uniform Redundant Carrier Placement for UW-OFDM,” Vehicular Technology Conf., Fall 2012, Quebec, Canada, Sep. 2012], for the case where no guard bands are present, and the maximum distance (MD) distribution, in case forbidden guard bands are present. We show in this paper that when the full band is available for redundant carriers, both the redundant carrier placement found with the B&B algorithm and the QU distribution result in a redundant energy that is very close to the theoretical lower bound. Although the B&B slightly outperforms the QU distribution, this implies that the QU distribution is a good low cost solution to the redundant carrier placement problem in the case of no guard bands. However, when there are forbidden guard bands in which the redundant carriers cannot be placed, the QU distribution can no longer be used. In that case, the results of the B&B algorithm are compared to that of the MD distribution. We show in this paper that the proposed B&B algorithm outperforms the MD distribution in the sense that it needs noticeably lower redundant energy.