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Conformal prediction provides a distribution-free coverage guarantee for predictive inference, yet its validity degrades under distribution shift, a common challenge in real-world deployment. We introduce DiffConf, a framework that uses score-based diffusion models as expressive priors over the data-generating process to enable adaptive conformal inference under temporal and covariate distribution shifts. The main idea is that the score function learned by a diffusion model encodes rich geometric information about the data manifold, which can be repurposed to construct nonconformity scores that are sensitive to distributional changes. We derive a diffusion-guided conformity score that integrates the learned score field with a lightweight online recalibration mechanism, providing finite-sample marginal coverage guarantees even when the data distribution evolves over time. Theoretically, we establish that DiffConf achieves asymptotic conditional coverage under mild regularity conditions on the drift rate, and we prove a regret bound that scales gracefully with the complexity of the distribution shift. Experiments on synthetic benchmarks, real-world tabular regression tasks, and high-dimensional image datasets demonstrate that DiffConf produces prediction sets that are simultaneously valid and more efficient than existing adaptive conformal methods, reducing average set size by 6--39% across the real-data benchmarks (and by over 50% under large synthetic mean shifts) while keeping coverage within one point of target.