Codec-based language models (LMs) have revolutionized text-to-speech (TTS). However, standard codecs entangle timbre and prosody, which hinders independent control in continuation-based LMs. To tackle this challenge, we propose DisCo-Speech, a zero-shot controllable TTS framework featuring a disentangled speech codec (DisCodec) and an LM-based generator. The core component DisCodec employs a two-stage design: 1) tri-factor disentanglement to separate speech into content, prosody, and timbre subspaces via parallel encoders and hybrid losses; and 2) fusion and reconstruction that merges content and prosody into unified content-prosody tokens suitable for LM prediction, while jointly optimizing reconstruction to address the disentanglement-reconstruction trade-off. This allows the LM to perform prosodic continuation from a style prompt while the decoder injects target timbre, enabling flexible zero-shot control. Experiments demonstrate that DisCo-Speech achieves competitive voice cloning and superior zero-shot prosody control. By resolving the core entanglement at the codec level, DisCo-Speech provides a robust foundation for controllable speech synthesis. Audio samples are available at: https://disco-speech.github.io/DisCo-demo/. Code and weights will be released at: https://github.com/disco-speech/DisCo-Speech upon acceptance.