After Nancy Hernandez gave birth to a baby girl in a hospital in Texas in 2013, she went to a county office to get a birth certificate, just as she had after her first two children were born in the state.

But officials told Ms. Hernandez, a Mexican immigrant living in the United States illegally, that the rules had changed. Without valid documents, she would not be able to get a birth certificate to show that her daughter was an American-born citizen.

Last year, Ms. Hernandez and about two dozen other immigrants sued, saying they could not obtain the documents Texas officials were demanding to prove their identities. On Friday, Texas agreed to a settlement that will expand the types of documents parents can present, allowing those without legal immigration status to obtain certificates for their children again.

The babies whose parents brought the federal suit were born in Texas medical facilities, so it was not in doubt that they were citizens. Lawyers for the parents said the settlement would be “life-changing” for them.