The data migration problem is the problem of computing an efficient plan for moving data stored on devices in a network from one configuration to another. Load balancing or changing usage patterns could necessitate such a rearrangement of data. In this paper, we consider the case where the objects are fixed-size and the network is complete. The direct migration problem is closely related to edge-coloring. However, because there are space constraints on the devices, the problem is more complex. Our main results are polynomial time algorithms for finding a near-optimal migration plan in the presence of space constraints when a certain number of additional nodes is available as temporary storage, and a 3/2-approximation for the case where data must be migrated directly to its destination.