Accurate identification of true versus false climate information in the digital age is critical. Misinformation can significantly affect public understanding and policymaking. Automated fact-checking seeks to validate claims against trustworthy factual data. This study tackles the challenge of fact-checking climate claims by leveraging the currently most capable Large Language Models (LLMs). To this end, we introduce Climinator, an acronym for CLImate Mediator for INformed Analysis and Transparent Objective Reasoning. It significantly boosts the performance of automated fact-checking by integrating authoritative, up-to-date sources within a novel debating framework. This framework provides a trustworthy and context-aware analysis incorporating multiple scientific viewpoints. Climinator helps identify misinformation in real time and facilitates informed dialog on climate change, highlighting AI's role in environmental discussions and policy with reliable data.