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If the change involves adding a father's information to a birth certificate on which the father's information was left blank, the supporting documentation needed is either the certified copy of a marriage license or an Acknowledgement of Paternity. However, if paternity was established by a court order, you must submit a certified copy of that court order. A court order will also be required if changes have previously been made to the birth certificate.
If the change you are wanting to make involves removing the father's information or removing one man's information and adding the information of another man, the supporting documentation must be a certified copy of a court order saying that the man listed on the birth certificate is not the biological father or that the man to be added to the birth certificate is the biological father. In some cases, a properly filled out and signed, first-time-ever Acknowledgement of Paternity can be submitted if ths "denial" portion has been filled out.
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Texas Vital Statistics ? Amending Paternity and Parentage on a Birth Certificate