You can only adopt a child that isn't biologically yours. If the child is biologically yours, then you're their legal parent and have no need to adopt.
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If the boyfriend and the biological mother were married, this could be accomplished by a step-parent adoption. However, since you are talking about your own boyfriend, this doesn't really sound like the case here. Being on the birth certificate isn't really all that helpful here as far as the adoption goes. There's no right to adopt a child against the biological mother's wishes just because the other parent has passed away.
There might be an option for parenting time based on his being on the birth certificate, but this would be much more likely if he has established a strong "parent-child relationship" with the child in the past 6 months. He'd have to consult directly with a family law attorney to learn more about this.
As a side note, the idea of him trying to force an adoption (or even parenting time, if he hasn't had frequent, substantial contact with the child...essentially forming the relationship that a parent would have) on the bio mom and child seems a bit distasteful. What I'm saying is, if the mom isn't on board with this, a court most likely won't be either.
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