This paper proposes an adaptive frequency offset scheme in an amplify-and-forward (AF) relay station (RS). When the RSs are at a cell edge, frequency resources are partially assigned between them and a base station (BS) to avoid inter-cell interference. On the other hand, the RS is expected to use a higher frequency band for transmission in the next generation mobile communication system. The whole frequency band can be used in the transmission between the RS and a mobile station (MS) while the RS uses only the restricted bandwidth in the low frequency band. In the proposed AF relay scheme, the frequency of the signal from the RS is shifted adaptively in accordance with the frequency response of the channel between the RS and the MS. In this scheme, the relay transmission is assumed to use a wider bandwidth between the RS and the MS than that between the BS and the RS. By making use of the better frequency response between the RS and the BS owing to the adaptive frequency offset scheme the total throughput of the cell improves. It is shown with numerical results obtained through computer simulation that for a two-user case the proposed scheme improves the total throughput by about 1.2 to 1.3 times with proportional fairness scheduling especially at a low Eb/N0 condition.