By right of law.
With the completion of the adoption proceedings, the Millers became the de jure as well as the de facto parents of the child.
Literally meaning "by right" in Latin, de jure is typically used in sentences where it is set in opposition to de facto. It is used with reference to things that have the force of law or operate under a right recognized by law. A de jure president is one duly elected under a nation's laws. A de facto ruler, on the other hand, may be exercising power that has been acquired through illegal means.