Assuming you were never married to the alleged father and absent a court order stating otherwise, at this time, you are the sole custodian of your child. However, if paternity is established, the alleged father will have to right to file a complaint for custody to establish his parental rights. In other words, by default, you have full custody of your child, but that could be subject to change through the court.
As for adoption, yes, it is possible for your husband to adopt your child through process called Step-Parent Adoption. Generally, whether or not the alleged father's consent is required is often the most important determination in these cases. If you were not married to the father, the alleged father did not sign the birth certificate (or paternity affidavit), parentage was not established through the child support enforcement agency, and the alleged father is not on the putative father registry, his consent may not be required. In addition, the alleged father’s consent may also not be required where he has had little (“de minimis”) to no contact with the child and has not provided support to the child for more than a one (1) year. However, in most cases, the alleged father's consent is required. In addition, the court will also look into the background of the adopting step-parent.
However, each case is very unique and fact specific. In any case, I highly recommend that you consult with an attorney as soon as possible to discuss your options.
If you and the previous father were not married then you already have full custody of your child. In Ohio a child born to unmarried parents is in the custody of the mother until a court enters a finding of custody of the father and grants rights to the father. Your new husband may be able to get a step-child adoption. There are certain things that will have to be done with serving the natural father and going through the punitive father registry. You should consult privately with an attorney who can assist you with the step-parent adoption. This is a very paperwork and fact specific area of law. Good luck.
Attorney Chris Beck
Beck Law Office, L.L.C.
The responses of Attorney Chris Beck to any questions posed on Avvo do NOT establish an Attorney-client relationship. Attorney Beck is available for private hire and consultation for a fee. Only after Attorney Beck is retained as counsel, or agrees to discuss this matter with you privately, shall he be legally deemed to be your Attorney. His responses herein are an attempt to assist persons temporarily based upon the very extremely limited amount of information provided by the questioner