Clarence Center is a hamlet and census-designated place in the town of Clarence in Erie County, New York, United States. The population was 2,257 at the 2010 census. It is part of the Buffalo–Niagara Falls Metropolitan Statistical Area.Clarence Center, within the south-central part of the town, is the location of most of the town government facilities, although the actual town hall is within the Clarence postal zone.Clarence Center is also one of the postal zones in the town with a ZIP code of 14032. This area is most of the north part of the town.