The combined input-crosspoint-queued (CICQ) crossbar switch is becoming attractive because of the simplicity of scheduling. Although the previously proposed round-robin algorithms achieves 100% throughput asymptotically under uniform Bernoulli i.i.d. traffic, these algorithms do not have a satisfactory performance under nonuniform traffic. In this paper, we propose a dual round-robin algorithm for a CICQ switch with one-cell crosspoint buffers. With our algorithm, each input arbiter is associated with dual round-robin pointers. Unlike the previously proposed algorithms, the new algorithm has distinctive roundrobin pointer updating rules which are efficient to cope with nonuniform traffic patterns. Extensive simulations show that our algorithm achieves nearly 100% throughput and low average delay under uniform and a broad class of nonuniform traffic patterns.