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601 4373050 AfterHours Services Available A Family Health Center in Port Gibson MS Make an appointment at Claiborne County Family Health Center in Port Gibson MS. We are a nonprofit organization that provides a variety of medical dental and public health services. The Centers trained staff is committed to fulfilling its mission To provide quality accessible available and affordable primary and preventive health care services to residents of Claiborne County and surrounding areas. We are committed to taking care of you our patients.