APIs play a pivotal role in modern software development by enabling seamless communication and integration between various systems, applications, and services. Component-based API synthesis is a form of program synthesis that constructs an API by assembling predefined components from a library. Existing API synthesis techniques typically implement dedicated search strategies over bounded spaces of possible implementations, which can be very large and time consuming to explore. In this paper, we present a novel approach of using large language models (LLMs) in API synthesis. LLMs offer a foundational technology to capture developer insights and provide an ideal framework for enabling more effective API synthesis. We perform an experimental evaluation of our approach using 135 real-world programming tasks, and compare it with FrAngel, a state-of-the-art API synthesis tool. The experimental results show that our approach completes 133 of the tasks, and overall outperforms FrAngel. We believe LLMs provide a very useful foundation for tackling the problem of API synthesis, in particular, and program synthesis, in general.