There has been a significant progress in Text-To-Speech (TTS) synthesis technology in recent years, thanks to the advancement in neural generative modeling. However, existing methods on any-speaker adaptive TTS have achieved unsatisfactory performance, due to their suboptimal accuracy in mimicking the target speakers' styles. In this work, we present Grad-StyleSpeech, which is an any-speaker adaptive TTS framework that is based on a diffusion model that can generate highly natural speech with extremely high similarity to target speakers' voice, given a few seconds of reference speech. Grad-StyleSpeech significantly outperforms recent speaker-adaptive TTS baselines on English benchmarks. Audio samples are available at https://nardien.github.io/grad-stylespeech-demo.