The emergence of audio language models is empowered by neural audio codecs, which establish critical mappings between continuous waveforms and discrete tokens compatible with language model paradigms. The evolutionary trends from multi-layer residual vector quantizer to single-layer quantizer are beneficial for language-autoregressive decoding. However, the capability to handle multi-domain audio signals through a single codebook remains constrained by inter-domain distribution discrepancies. In this work, we introduce UniCodec, a unified audio codec with a single codebook to support multi-domain audio data, including speech, music, and sound. To achieve this, we propose a partitioned domain-adaptive codebook method based on domain Mixture-of-Experts strategy to capture the distinct characteristics of each audio domain. Furthermore, to enrich the semantic density of the codec without auxiliary modules, we propose a self-supervised mask prediction modeling approach. Comprehensive objective and subjective evaluations demonstrate that UniCodec achieves excellent audio reconstruction performance across the three audio domains, outperforming existing unified neural codecs with a single codebook, and even surpasses state-of-the-art domain-specific codecs on both acoustic and semantic representation capabilities.