You and/or you and your partner can adopt your sister's daughter if she consents. Find a good adoption attorney to help you with the process.
There are a couple of hurdles you have to get over before you can adopt. One is you need her consent but you also need the consent of the birth father - your first bet is to apply for temporary custody of the infant. In some states adoptions cannot happen until the child is three months old which gives you time to get all your information in place. The other hurdle is whether or not you will need a home study to insure that you and your partner are "qualified" - in NJ we do background checks and a home study. Realistically you are young yourselves and stability is objective. You would do well to consult personally with an attorney who is familiar with adoptions in NYC. Best wishes to you both.
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As the other attorneys have suggested, you really should speak with an experienced adoption attorney. There are several in New York. I recommend you go to adoptionattorneys.org and contact one of the Fellows of the American Academy of Adoption Attorneys in your area. Also, you did not indicate if your sister also lives in New York. If not, then you should probably speak with an adoption attorney in her state of residence as well since most adoption law is state specific. Best of luck to you and your partner.
Please note this answer does not create an attorney-client relationship or provide specific legal advice. It is intended for general information only.
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