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Thank you for the additional information. It is clear that the father wants nothing to do with the child. As such, the decision to adopt the child and have them as your own, is up to you. You could continue to live with her and care for the child and when the time comes that you marry, you could then adopt the child. If the father is willing to terminate his rights and anything related to that, the court should not have a problem with this happening. At this point, he nor the mother has to do anything, since he is not listed on the birth certificate and there is nothing right now for him to sign away but when the time comes that you want to take the next step, you can and adopt the child.
He can relinquish his rights to the child, once it is established that he is the biological father and that would prevent him from coming back later. Then, once married, you could adopt the child.