In frequency duplex division systems, it is very challenging to position the receiver (Rx) that belongs to the third-party system since the anchors usually do not know which frequency band the Rx uses for transmission. As a result, they can only use the received signal from the third-party transmitter (Tx) to position the Rx. In this paper, we propose a relay-based positioning method to estimate the location of the third-party Rx. In our method, we let each anchor alternatively act as a full-duplex amplify-and-forward (AF) relay for the Rx, which artificially creates a relay path. By estimating the time-difference-of-arrival (TDOA) between the direct and relay paths, the anchors can estimate the Rx location based on the received signal from the third-party Tx. Simulation results indicate the effectiveness of the proposed method.