what do they do in order to sign over rights to a child?
Lawsuit / Dispute Attorney
Unless there is someone stepping in to adopt the child, the father may be able to terminate his rights but not his obligation for child support
Divorce / Separation Lawyer
Before an order terminating a parents rights may be granted, the judge must find such an order to be in the child's best interest. In my experience, most judges do not find a termination of parental rights to be in a child's best interest in such a circumstance unless there is another man standing willing to adopt and support the child.