Background noise considerably reduces the accuracy and reliability of speaker verification (SV) systems. These challenges can be addressed using a speech enhancement system as a front-end module. Recently, diffusion probabilistic models (DPMs) have exhibited remarkable noise-compensation capabilities in the speech enhancement domain. Building on this success, we propose Diff-SV, a noise-robust SV framework that leverages DPM. Diff-SV unifies a DPM-based speech enhancement system with a speaker embedding extractor, and yields a discriminative and noise-tolerable speaker representation through a hierarchical structure. The proposed model was evaluated under both in-domain and out-of-domain noisy conditions using the VoxCeleb1 test set, an external noise source, and the VOiCES corpus. The obtained experimental results demonstrate that Diff-SV achieves state-of-the-art performance, outperforming recently proposed noise-robust SV systems.