Recommender systems play a vital role in various online services. However, the insulated nature of training and deploying separately within a specific domain limits their access to open-world knowledge. Recently, the emergence of large language models (LLMs) has shown promise in bridging this gap by encoding extensive world knowledge and demonstrating reasoning capability. Nevertheless, previous attempts to directly use LLMs as recommenders have not achieved satisfactory results. In this work, we propose an Open-World Knowledge Augmented Recommendation Framework with Large Language Models, dubbed KAR, to acquire two types of external knowledge from LLMs -- the reasoning knowledge on user preferences and the factual knowledge on items. We introduce factorization prompting to elicit accurate reasoning on user preferences. The generated reasoning and factual knowledge are effectively transformed and condensed into augmented vectors by a hybrid-expert adaptor in order to be compatible with the recommendation task. The obtained vectors can then be directly used to enhance the performance of any recommendation model. We also ensure efficient inference by preprocessing and prestoring the knowledge from the LLM. Extensive experiments show that KAR significantly outperforms the state-of-the-art baselines and is compatible with a wide range of recommendation algorithms. We deploy KAR to Huawei's news and music recommendation platforms and gain a 7\% and 1.7\% improvement in the online A/B test, respectively.