Grossglauser and Tse (2001) introduced a mobile random network model where each node moves independently on a unit disk according to a stationary uniform distribution and showed that a throughput of Theta(1) is achievable. El Gamal, Mammen, Prabhakar, and Shah (2004) showed that the delay associated with this throughput scales as Theta(nlogn), when each node moves according to an independent random walk. In a later work, Diggavi, Grossglauser, and Tse (2002) considered a random network on a sphere with a restricted mobility model, where each node moves along a randomly chosen great circle on the unit sphere. They showed that even with this one-dimensional restriction on mobility, constant throughput scaling is achievable. Thus, this particular mobility restriction does not affect the throughput scaling. This raises the question whether this mobility restriction affects the delay scaling. This correspondence studies the delay scaling at Theta(1) throughput for a random network with restricted mobility. First, a variant of the scheme presented by Diggavi, Grossglauser, and Tse (2002) is presented and it is shown to achieve Theta(1) throughput using different (and perhaps simpler) techniques. The exact order of delay scaling for this scheme is determined, somewhat surprisingly, to be of Theta(nlogn), which is the same as that without the mobility restriction. Thus, this particular mobility restriction does not affect either the maximal throughput scaling or the corresponding delay scaling of the network. This happens because under this 1-D restriction, each node is in the proximity of every other node in essentially the same manner as without this restriction