We consider a network where an infection cascade has taken place and a subset of infected nodes has been partially observed. Our goal is to reconstruct the underlying cascade that is likely to have generated these observations. We reduce this cascade-reconstruction problem to computing the marginal probability that a node is infected given the partial observations, which is a #P-hard problem. To circumvent this issue, we resort to estimating infection probabilities by generating a sample of probable cascades, which span the nodes that have already been observed to be infected, and avoid the nodes that have been observed to be uninfected. The sampling problem corresponds to sampling directed Steiner trees with a given set of terminals, which is a problem of independent interest and has received limited attention in the literature. For the latter problem we propose two novel algorithms with provable guarantees on the sampling distribution of the returned Steiner trees. The resulting method improves over state-of-the-art approaches that often make explicit assumptions about the infection-propagation model, or require additional parameters. Our method provides a more robust approach to the cascadereconstruction problem, which makes weaker assumptions about the infection model, requires fewer additional parameters, and can be used to estimate node infection probabilities. Empirically, we validate the proposed reconstruction algorithm on real-world graphs with both synthetic and real cascades. We show that our method outperforms all other baseline strategies in most cases