Rockport is a city in Aransas County, Texas, United States. The population was 7,385 at the 2000 census. It is the county seat of Aransas County. The town is named for the rock ledge underlying its shore. It is located south of the Live Oak Peninsula, which stretches out into Copano Bay. Rockport apartments for rent comparison page shows most up to date rent information on Rockport apartments and houses. Rent information is obtained from multiple rental sources but we cannot guarantee their accuracy.