It will be difficult for the step-parent to adopt the child without the consent of the biological father or if the father appears in the proceeding and contests the adoption. You can petition in Family Court or Surrogates Court and the biological father will have to be given notice. If the child is adopted this will terminate the biological parents parental rights such as visitation and the obligation to pay support and he can take the step-father's last name. What you're seeking to do is possible, but tricky, depending on how the biological father reacts and his desires. Retaining a Family Court lawyer familiar with adoption procedures would be very helpful here.
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I respectfully disagree with the other poster - on the facts you've presented, it appears you clearly make a cause of action for abandonment, which is the most common ground for a step-parent adoption. Thus, I encourage to you follow-up with a Family Law attorney in your area, particularly one with experience in adoption cases.
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