Hunter Biden is the "biological and legal father" of Baby Doe, Circuit Judge Holly Meyer said in an order signed Tuesday.
DNA results showed "with near scientific certainty" that he fathered the child, she noted.
She also ordered the Arkansas Department of Health to issue a "new or substituted" birth certificate listing Robert Hunter Biden as the father and Lunden Alexis Roberts as the mother of the child, born in Aug. 2018.
Originally referred to as "Baby Doe," Biden's child will henceforth be referred to in court documents by his or her initials: NJR. The court had previously ruled that the child's actual name would be sealed.
In a separate order, the judge awarded "primary physical and legal custody" of the child to Roberts.
As the non-custodial parent, Biden "shall have visitation with the child as agreed between the parties," Meyer wrote in the Order on Custody and Visitation, which was also signed and filed Tuesday afternoon.
Read Wednesday's Arkansas Democrat-Gazette for full details.