In addition to the traditional B1I signal, the third-generation Beidou navigation satellite system (BDS-3) broadcasts a new civilian signal, namely, B1C. These two signals can be combined to form an asymmetric double-sideband signal, which is called the B1 composite signal. The Gabor bandwidth of the B1 composite signal is much larger than that of B1I and B1C signals, which indicates that it has much better ranging performance in theory. Traditional tracking methods for the B1 composite signal treat it as a whole, which can be referred to as wideband tracking methods. However, due to the large bandwidth of the B1 composite signal, the sampling rate and processing rate required by these wideband tracking methods are both very high, resulting in high computational complexity and hardware cost. In order to solve this problem, this paper proposes a computationally efficient tracking method for the B1 composite signal. The proposed method first separates the B1 composite signal into two components: B1I and B1C signals, so that they can be down-sampled to reduce the processing rate required by the correlator channel. Then the down-sampled B1I and B1C signals are jointly processed in the tracking loop to obtain high-precision ranging measurements. Theoretical analysis and experimental results illustrate that the proposed method can achieve high ranging performance with low computational complexity.