Mixed reality applications require tracking the user's full-body motion to enable an immersive experience. However, typical head-mounted devices can only track head and hand movements, leading to a limited reconstruction of full-body motion due to variability in lower body configurations. We propose BoDiffusion - a generative diffusion model for motion synthesis to tackle this under-constrained reconstruction problem. We present a time and space conditioning scheme that allows BoDiffusion to leverage sparse tracking inputs while generating smooth and realistic full-body motion sequences. To the best of our knowledge, this is the first approach that uses the reverse diffusion process to model full-body tracking as a conditional sequence generation task. We conduct experiments on the large-scale motion-capture dataset AMASS and show that our approach outperforms the state-of-the-art approaches by a significant margin in terms of full-body motion realism and joint reconstruction error.