Bundle recommendation aims to recommend a set of items to each user. However, the sparser interactions between users and bundles raise a big challenge, especially in cold-start scenarios. Traditional collaborative filtering methods do not work well for this kind of problem because these models rely on interactions to update the latent embedding, which is hard to work in a cold-start setting. We propose a new approach (DisCo), which relies on a personalized Diffusion backbone, enhanced by disentangled aspects for the user's interest, to generate a bundle in distribution space for each user to tackle the cold-start challenge. During the training phase, DisCo adjusts an additional objective loss term to avoid bias, a prevalent issue while using the generative model for top-$K$ recommendation purposes. Our empirical experiments show that DisCo outperforms five comparative baselines by a large margin on three real-world datasets. Thereby, this study devises a promising framework and essential viewpoints in cold-start recommendation. Our materials for reproducibility are available at: https://github.com/bt-nghia/DisCo.