The hospital has no role here at all. They simply have people fill out the paperwork. The mother is automatically named as the one who gives birth. Then each state has their own laws in place to govern the procedure for issuing an amended certificate of live birth. Some required the new mom to go through a legal adoption procedure which involves a voluntary termination of the birth mother's rights. Other allow that process to court approved prior to the birth and then a simple form is submitted to issue an amended certificate.
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