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In order to do this, you would likely have to first marry your girlfriend in order to be the mother's spouse and stepfather to the child. Then, you would need to file a petition for adoption and ask the court to issue a summons to the father (known or unknown) and then get a court order approving the adoption and amending the birth certificate to name you as the father.
This certainly can be done, but you must follow the proper procedure and get a court order to make things official and legal.