The answer is yes and no.  If biological mom has full custody and has the right to determine the residence of the child, then the mother has the right to allow the child to live with the step-parent with no "up front" legal issues.  However, if biological father does not like this arrangement, then he can always file a motion with the court to object to her decision and ask the court to award him full custody since mom appears to be delegating her parental duties to someone else. So to answer your question.... it is possible for mom to consent, but by doing so, she could face a potential motion to modify.