Yes. It is possible you can be granted sole custody. An unemployed father is not necessarily the best person for a child to live with. You need to consult with an attorney who specializes in custody issues and can review your situation and explain your options.
In the US an unmarried mother has sole custody until the court has awarded custody rights to the father. It is the father who has to prove to the court that the child would be better off with him. If he is unemployed that is not likely.
General information regarding custody if the parties are not married:
Generally, if the parents are unmarried the mother has sole custody and control in most states until the father can establish his paternity. Remember, a child's mother can always be identified by medical records. Since the father didn't give birth and he was not legally married at the time of the birth he must establish his paternity by signing the birth certificate at the time of birth (waiving DNA testing rights) which must be done with mother's consent.
If he doesn't sign the birth certificate then he must seek another way to establish his paternity and that is done through a DNA test. A paternity test can be arranged through the court. Once paternity is established in court, the father can request visitations or custody through the court. If the mother retains physical custody she can request that the court issue a child support order. If the father gets physical custody he can request a child support order.
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