A certificate must be prepared for a child born out of wedlock in this State to include the name of the father upon receipt of a sworn acknowledgment of paternity signed by both parents to include the surname by which the child is to be known. However, if the reputed father or the mother is deceased, if another man is shown as the father of the child on the original birth certificate or if the original birth certificate states that the mother was married, a new certificate may be prepared only when a determination of paternity is made by the Family Court pursuant to Section 20-7-952. A paternity acknowledgment must be provided to the State Department of Social Services from the appropriate state agency upon request at no charge for the purpose of establishing a child support obligation and otherwise a paternity acknowledgment is not subject to inspection except upon order of the Family Court.