Recent advancements in diffusion models have shown promising results in sequential recommendation (SR). However, current diffusion-based methods still exhibit two key limitations. First, they implicitly model the diffusion process for target item embeddings rather than the discrete target item itself, leading to inconsistency in the recommendation process. Second, existing methods rely on either implicit or explicit conditional diffusion models, limiting their ability to fully capture the context of user behavior and leading to less robust target item embeddings. In this paper, we propose the Dual Conditional Diffusion Models for Sequential Recommendation (DCRec), introducing a discrete-to-continuous sequential recommendation diffusion framework. Our framework introduces a complete Markov chain to model the transition from the reversed target item representation to the discrete item index, bridging the discrete and continuous item spaces for diffusion models and ensuring consistency with the diffusion framework. Building on this framework, we present the Dual Conditional Diffusion Transformer (DCDT) that incorporates the implicit conditional and the explicit conditional for diffusion-based SR. Extensive experiments on public benchmark datasets demonstrate that DCRec outperforms state-of-the-art methods.