Unfortunately in California she can't give up her parental rights unless there's someone else ready to pick them up.
You CAN enter into an agreement with her acknowledging that you are his father, and giving you sole legal and sole physical custody rights.
Then file a Petition to Establish Parental Relationship asserting your parentage, and requesting amendment of the birth certificate and sole legal/physical custody. You would be able to use the DNA test to confirm your parentage and the agreement to support your requests.
Sign up to receive a 3-part series of useful information and advice about child custody law.