To infer the structure of a diffusion network from observed diffusion results, existing approaches customarily assume that observed data are complete and contain the final infection status of each node, as well as precise timestamps of node infections. Due to high cost and uncertainties in the monitoring of node infections, exact timestamps are often unavailable in practice, and even the final infection statuses of nodes are sometimes missing. In this work, we study how to carry out diffusion network inference without infection timestamps, using only partial observations of the final infection statuses of nodes. To this end, we iteratively infer the structure of the target diffusion network with observed data and imputed values for missing data, and learn the most likely infection transmission probabilities between nodes w.r.t. current inferred structure, which then help us update the imputation of missing data in turn. Extensive experimental results on both synthetic and real-world networks show that our approach can properly handle missing data and accurately uncover diffusion network structures.