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Can I automatically align the border of one shapefile to overlap the border of another, without editing each vertice individually?

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I believe what you are searching for is called Integrate (ArcGIS Help 10.2 - Integrate (Data Management)):

Integrate is used to maintain the integrity of shared feature boundaries by making features coincident if they fall within the specified x,y tolerance. Features that fall within the specified x,y tolerance are considered identical or coincident.

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There are many methods available and they all have pluses and minuses. See http://resources.arcgis.com/en/help/main/10.2/index.html#//01m800000015000000 for some but not all. The advantage of the Integrate tool (or Validation of two layers in a topology) is that it happens with little work, but only if the differences are within the tolerance. You can increase the tolerance but at some point it may move vertices in both layers you don't want moved. Since both change the inputs do them on copies or in an edit session in case you don't like the results. Note, even though you can set a priority for which layer is to align to the other in both; people on these forums have often commented that both layers had vertices moved. I usually end up either 1.) choosing to reshape one polygon and using the trace tool to match it to the other layer, or 2.) (since I have ArcGIS Advanced) using the desired shape feature to first split the target then deleting the "scrap", and then using construct features taking into account existing ones to fill in the gaps, then merging them with the main poly to finish.

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