How is each region of the DNA replicated only once?
DNA Replication ensures it only occurs once by the use of licensing factors. These factors are released and bind to the origins of replication during one distinct phase. After the factors have been released and the first phase has ended, the actual replication initiates. 

Information on the licensing factors:

The licensing factors used to create the pre-replication complex is the Minichromosome Maintenance (Mcm2-7) protein, Cdc6, and Cdt1. The Cdc6 and Cdt1 are required to load the Mcm’s onto the DNA and therefore, are highly controlled during the cell cycle. The Mcm2-7 hexamer is the actual DNA helicase that is used during DNA replication. 

The way that the re-replication of DNA is prevented is that when the DNA replication starts, the Mcm helicase moves away from the ORC and the newly replicated DNA, meaning it then cannot be reinitiated. Along with the physical moving of the helicase, Cdt1’s activity is suppressed by the protein Geminin, which prevents the licensing of the helicase during times of non-replication. 

Image from the article on the licensing.


More information on this topic:
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2688777/#!po=21.8085