Large Language Models (LLMs) exhibit remarkable human-like predictive capabilities. However, it is challenging to deploy LLMs to provide efficient and adaptive inference services at the edge. This paper proposes a novel Cloud-Edge Collaboration framework for LLMs (CE-CoLLM) to tackle these challenges. First, we identify the transmission of LLM contextual data between the cloud and edge as a key performance bottleneck, which introduces substantial communication overhead that dominates overall inference latency and makes naïve cloud-edge collaboration for LLMs inefficient. Second, we introduce a suite of novel techniques, including a latency-aware early exit mechanism and efficient cloud context management, into CE-CoLLM, which collectively reduce communication overhead and preserve LLM inference accuracy. Third, we design two adaptive inference modes to accommodate diverse edge environments: (1) a low-latency standalone edge inference mode that enables reliable edge-side independent LLM inference even under unstable network conditions, and (2) a high-accuracy cloud-edge collaborative inference mode that adaptively leverages cloud resources to enhance prediction accuracy. Extensive experiments on multiple benchmark datasets demonstrate that CE-CoLLM reduces overall inference time by up to 13.81% and offloads over 84.53% of the computational workload from the cloud to the edge, compared to conventional cloud-based LLM deployment, without sacrificing prediction accuracy. The code is provided on GitHub at https://github.com/mlsysx/CE-CoLLM.