Sole custody doesn't exist anymore. It's sole parental responsibility.
Can you get sole parental responsibility? Under the circumstances you've provided, HIGHLY unlikely. A child has a right to know and be cared for by both parents on a regular basis, so a mother wanting to deny a child of that right by giving the child up for adoption when the father does not consent to the child being put up for adoption could lead to the court just granting sole parental responsibility to the father.
If you don't mind giving some more insight, why does the mother want to give the child up for adoption instead of just giving parental responsibility to the father?