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We propose UnitSpeech, a speaker-adaptive speech synthesis method that fine-tunes a diffusion-based text-to-speech (TTS) model using minimal untranscribed data. To achieve this, we use the self-supervised unit representation as a pseudo transcript and integrate the unit encoder into the pre-trained TTS model. We train the unit encoder to provide speech content to the diffusion-based decoder and then fine-tune the decoder for speaker adaptation to the reference speaker using a single $<$unit, speech$>$ pair. UnitSpeech performs speech synthesis tasks such as TTS and voice conversion (VC) in a personalized manner without requiring model re-training for each task. UnitSpeech achieves comparable and superior results on personalized TTS and any-to-any VC tasks compared to previous baselines. Our model also shows widespread adaptive performance on real-world data and other tasks that use a unit sequence as input.