The Opinion is here.
In a 5-to-4 vote the Supreme Court decided, as Justice Kennedy put it:
These considerations lead to the conclusion that the right to marry is a fundamental right inherent in the liberty of the person, and under the Due Process and Equal Protection Clauses of the Fourteenth Amendment couples of the same-sex may not be deprived of that right and that liberty. The Court now holds that same-sex couples may exercise the fundamental right to marry. No longer may this liberty be denied to them. Baker v. Nelson must be and now is overruled, and the State laws challenged by Petitioners in these cases are now held invalid to the extent they exclude same-sex couples from civil marriage on the same terms and conditions as opposite sex couples (23–24 OBERGEFELL ET AL. v. HODGES, DIRECTOR, OHIO DEPARTMENT OF HEALTH, ET AL).
No union is more profound than marriage, for it embodies the highest ideals of love, fidelity, devotion, sacrifice, and family. In forming a marital union, two people become something greater than once they were. As some of the petitioners in these cases demonstrate, marriage embodies a love that may endure even past death. It would misunderstand these men and women to say they disrespect the idea of marriage. Their plea is that they do respect it , respect it so deeply that they seek to find its fulfillment for themselves. Their hope is not to be condemned to live in loneliness, excluded from one of civilization’s oldest institutions. They ask for equal dignity in the eyes of the law. The Constitution grants them that right (p. 28 OBERGEFELL ET AL. v. HODGES, DIRECTOR, OHIO DEPARTMENT OF HEALTH, ET AL).
In other words, there is no more same-sex marriage; there’s just marriage.