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The prospective adopting parents simply need to file a petition to adopt this child. You, as the biological parent, can sign a joinder and waiver of process for the petition, showing the court that you don't oppose the prayer for adoption.
At that point, the court holds a hearing and decides whether it is in the child's best interest to be adopted.
The fact that one of the people has a felony conviction is certainly not going to help anything. But, if this occurred years ago and no further criminal activity has occurred, and if this was not an abuse or sex crime, it should not be a reason for the adoption not to go through.
My suggestion is to speak to a family law attorney in your area about this and he/she can assist in preparing and filing the petition, plus the court appearance.