The conventional divide-and-conquer algorithm for constructing Voronoi diagrams is revised into a numerically robust one. The strategy for the revision is a topology-oriented approach. That is, at every step of the algorithm, consistency of the topological structure is considered more important than the result of numerical computation, and hence numerical values are used only when they do not contradict the topological structure. The resultant new algorithm is completely robust in the sense that, no matter how poor the precision may be, the algorithm always carries out its task, ending up with a topologically consistent output, and is correct in the sense that the output "converges" to the true Voronoi diagram as the precision becomes higher. Moreover, it is efficient in the sense that it achieves the same time complexity as the original divide-and-conquer algorithm unless the precision in computation is too poor. The performance of the algorithm is also verified by computational experiments.