Automatic speech recognition (ASR) has been significantly improved in the past years. However, most robust ASR systems are based on air-conducted (AC) speech, and their performances in low signal-to-noise-ratio (SNR) conditions are not satisfactory. Bone-conducted (BC) speech is intrinsically insensitive to environmental noise, and therefore can be used as an auxiliary source for improving the performance of an ASR at low SNR. In this paper, we first develop a multi-modal Mandarin corpus, which contains air- and bone-conducted synchronized speech (ABCS). The multi-modal speeches are recorded with a headset equipped with both AC and BC microphones. To our knowledge, it is by far the largest corpus for conducting bone conduction ASR research. Then, we propose a multi-modal conformer ASR system based on a novel multi-modal transducer (MMT). The proposed system extracts semantic embeddings from the AC and BC speech signals by a conformer-based encoder and a transformer-based truncated decoder. The semantic embeddings of the two speech sources are fused dynamically with adaptive weights by the MMT module. Experimental results demonstrate the proposed multi-modal system outperforms single-modal systems with either AC or BC modality and multi-modal baseline system by a large margin at various SNR levels. It also shows the two modalities complement with each other, and our method can effectively utilize the complementary information of different sources.