(a) A new certificate of birth shall be prepared by the State Registrar of Vital Records for a child born out of wedlock in this state upon receipt of a:

1. (1) Sworn acknowledgement of paternity signed by both parents; and
2. (2) Written request by both parents that the child's surname be changed on the certificate.

(b) However, if another man is shown as the father of the child on the original certificate, a new certificate may be prepared only:

1. (1) When a determination of paternity is made by a court of competent jurisdiction; or
2. (2) Following adoption.