Controllable layout generation refers to the process of creating a plausible visual arrangement of elements within a graphic design (e.g., document and web designs) with constraints representing design intentions. Although recent diffusion-based models have achieved state-of-the-art FID scores, they tend to exhibit more pronounced misalignment compared to earlier transformer-based models. In this work, we propose the LAyout Constraint diffusion modEl (LACE), a unified model to handle a broad range of layout generation tasks, such as arranging elements with specified attributes and refining or completing a coarse layout design. The model is based on continuous diffusion models. Compared with existing methods that use discrete diffusion models, continuous state-space design can enable the incorporation of continuous aesthetic constraint functions in training more naturally. For conditional generation, we propose injecting layout conditions in the form of masks or gradient guidance during inference. Empirical results show that LACE produces high-quality layouts and outperforms existing state-of-the-art baselines. We will release our source code and model checkpoints.