Generative diffusion is a machine learning technique to generate high-quality samples from complex data distributions. Much of its success can be attributed to the recently developed techniques that flexibly control the data generation process, without additional training effort. These methods control a pre-trained diffusion model towards specific regions of interest, which are determined by external information such as class labels, masks, or text descriptions. However, these approaches are typically based on heuristic guidance techniques and break the consistency on which the theoretical justification of generative diffusion relies. This is problematic when applying these controlled data generation techniques to tasks that are sensitive to distribution characteristics rather than the perceptual quality of individual samples. To this end, we introduce an asymptotically consistent approach for conditioning generative diffusion models without retraining the entire system. We use an importance sampling technique for simulating diffusion bridges, where multiple draws of a guided proposal process are reweighted to resemble paths of the true conditioned denoising process. A theoretical analysis shows that under certain assumptions, our approach has a vanishing error. In an empirical analysis, we find that specific nuances to the performance trade-off appear with a finite amount of computational effort. Specifically, the effectiveness of our approach highly depends on the choice of the proposal process and the allocation of computational effort towards independent runs of our algorithm.