We consider a network design problem (NDP) under random demand with unknown distribution for which only a small number of observations are known. We design arc capacities in the first stage and optimize single-commodity network flows after realizing the demand in the second stage. The objective is to minimize the total cost of allocating arc capacities, flowing commodities, and penalty for unmet demand. We formulate a distributionally robust NDP (DR-NDP) by constructing an ambiguity set of the unknown demand distribution based on marginal moment information, to minimize the worst-case total cost over all possible distributions. Approximating polynomials with piecewise-linear functions, we reformulate DR-NDP as a mixed-integer linear program optimized via a cutting-plane algorithm. We test diverse network instances to compare DR-NDP with a stochastic programming approach, a deterministic benchmark model, and a robust NDP formulation. Our results demonstrate adequate robustness of optimal DR-NDP solutions and how they perform under varying demand, modeling parameter, network, and cost settings. The results highlight potential niche uses of DR-NDP in data-scarce contexts.