Linear sand dunes--dunes that extend parallel to each other rather than in star-like or crescentic forms--are the most abundant type of desert sand dune. But because their development and their internal structure are poorly understood, they are rarely recognized in the rock record. Models of linear dune development have not been able to take into account the sub-surface structure of existing dunes, but have relied instead either on the extrapolation of short-term measurements of winds and sediment transport or on observations of near-surface internal sedimentary structures. From such studies, it has not been clear if linear dunes can migrate laterally. Here we present images produced by ground penetrating radar showing the three-dimensional sedimentary structure of a linear dune in the Namib sand sea, where some of the world's largest linear dunes are situated. These profiles show clear evidence for lateral migration in a linear dune. Moreover, the migration of a sinuous crest-line along the dune produces divergent sets of cross-stratification, which can become stacked as the dune height increases, and large linear dunes can support superimposed dunes that produce stacked sets of trough cross-stratification. These clear structural signatures of linear dunes should facilitate their recognition in geological records.