Existing approaches for all-in-one weather-degraded image restoration suffer from inefficiencies in leveraging degradation-aware priors, resulting in sub-optimal performance in adapting to different weather conditions. To this end, we develop an adaptive degradation-aware self-prompting model (ADSM) for all-in-one weather-degraded image restoration. Specifically, our model employs the contrastive language-image pre-training model (CLIP) to facilitate the training of our proposed latent prompt generators (LPGs), which represent three types of latent prompts to characterize the degradation type, degradation property and image caption. Moreover, we integrate the acquired degradation-aware prompts into the time embedding of diffusion model to improve degradation perception. Meanwhile, we employ the latent caption prompt to guide the reverse sampling process using the cross-attention mechanism, thereby guiding the accurate image reconstruction. Furthermore, to accelerate the reverse sampling procedure of diffusion model and address the limitations of frequency perception, we introduce a wavelet-oriented noise estimating network (WNE-Net). Extensive experiments conducted on eight publicly available datasets demonstrate the effectiveness of our proposed approach in both task-specific and all-in-one applications.