Machine learning as a service has been widely deployed to utilize deep neural network models to provide prediction services. However, this raises privacy concerns since clients need to send sensitive information to servers. In this paper, we focus on the scenario where clients want to classify private images with a convolutional neural network model hosted in the server, while both parties keep their data private. We present FALCON, a fast and secure approach for CNN predictions based on Fourier Transform. Our solution enables linear layers of a CNN model to be evaluated simply and efficiently with fully homomorphic encryption. We also introduce the first efficient and privacy-preserving protocol for softmax function, which is an indispensable component in CNNs and has not yet been evaluated in previous works due to its high complexity. We implemented the FALCON and evaluated the performance on real-world CNN models. The experimental results show that FALCON outperforms the best known works in both computation and communication cost.