The learning-based video compression method has made significant progress in recent years, exhibiting promising compression performance compared with traditional video codecs. However, prior works have primarily focused on advanced compression architectures while neglecting the rate control technique. Rate control can precisely control the coding bitrate with optimal compression performance, which is a critical technique in practical deployment. To address this issue, we present a fully neural network-based rate control system for learned video compression methods. Our system accurately encodes videos at a given bitrate while enhancing the rate-distortion performance. Specifically, we first design a rate allocation model to assign optimal bitrates to each frame based on their varying spatial and temporal characteristics. Then, we propose a deep learning-based rate implementation network to perform the rate-parameter mapping, precisely predicting coding parameters for a given rate. Our proposed rate control system can be easily integrated into existing learning-based video compression methods. The extensive experimental results show that the proposed method achieves accurate rate control on several baseline methods while also improving overall rate-distortion performance.