Text watermarking for large language models (LLMs) enables model owners to verify text origin and protect intellectual property. While watermarking methods for closed-source LLMs are relatively mature, extending them to open-source models remains challenging, as developers cannot control the decoding process. Consequently, owners of open-source LLMs lack practical means to verify whether text was generated by their models. A core difficulty lies in embedding watermarks directly into model weights without hurting detectability. A promising idea is to distill watermarks from a closed-source model into an open one, but this suffers from (i) poor detectability due to mismatch between learned and predefined patterns, and (ii) fragility to downstream modifications such as fine-tuning or model merging. To overcome these limitations, we propose PRO, a Precise and Robust text watermarking method for open-source LLMs. PRO jointly trains a watermark policy model with the LLM, producing patterns that are easier for the model to learn and more consistent with detection criteria. A regularization term further simulates downstream perturbations and penalizes degradation in watermark detectability, ensuring robustness under model edits. Experiments on open-source LLMs (e.g., LLaMA-3.2, LLaMA-3, Phi-2) show that PRO substantially improves both watermark detectability and resilience to model modifications.