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She and her husband, hedge fund manager Marc Mezvinsky, announced the news on Saturday morning, naming their child Charlotte Clinton Mezvinsky.
Writing on Twitter, Ms Clinton, 34, said they were “full of love, awe and gratitude”.
Mr Clinton, the former US President, and his wife, who is deliberating whether to run for office in the White House in 2016, had spoken of their excitement leading up to the birth.
The former Secretary of State, Mrs Clinton, called being a grandmother her “most exciting title yet” and had delayed decisions about her campaign until the baby’s arrival.