You do not have to adopt the girl and no one can force you to. However, if you already have custody and her parents are on board, the cost should be fairly nominal.
This answer is for general informational purposes only. This attorney does not intend to give legal advice and no attorney client relationship is created. If you desire legal advice you should seek a lawyer licensed in your state.
Your adoption should be relatively uncomplicated based upon your brief description. I'm assuming that the girl is agreeable, that you have a "permanent custody order" from a juvenile court, the natural mother will sign a consent to adoption, and the father has already surrendered his parental rights probably in the juvenile court. All these facts must be documented with the probate court hearing the adoption request. Your home will be subject to a home study for suitability/premises and a criminal background check must be performed on anyone residing therein over 18 years of age. Should either your husband or yourself have any prior marriages/divorces, they must be documented, as well as your marriage certificate. The court investigator will interview both of you. Procedures vary depending upon your relationship to the child, whether you are step-parent, grandparent or non-relative, as well as the cost in court fees. These also vary depending upon the county in which you live. Attorney fees will depend upon the specifics of all of the above, before a quote can be made. Attorney fees are subject to approval of the court. All parties must appear before the court at the final hearing. Other documentation, questionnaires etc. will be required, and the process usually takes several months. I hope this has been beneficial.