The high court's similar orders in two challenges to the ban mean that most travelers from these six countries cannot enter the United States while the cases are ongoing..
Lower courts had exempted more travelers who they said had "bona fide" connections to the United States, such as the grandparents and in-laws of citizens. Those exemptions mirrored ones designed by the Supreme Court in June, when the justices allowed only part of an earlier, temporary travel ban. The new ban, issued in September, is of indefinite length.
The orders are a victory for Trump and Republicans who have supported the ban.
"Now that the Supreme Court has ordered the lifting of restrictions on this ban, minimum security standards for entry into the United States can be enforced," said Michael Glassner, executive director of Donald J. Trump for President.
By allowing the full travel ban to take effect for now, the justices may be insinuating that they are likely to uphold it on the merits at a later date -- an interpretation the American Civil Liberties Union (ACLU) disputed.
"It's unfortunate that the full ban can move forward for now, but this order does not address the merits of our claims," said Omar Jadwat, who directs the ACLU's immigrants' rights project. "We continue to stand for freedom, equality, and for those who are unfairly being separated from their loved ones."
The unsigned orders issued Monday urge two federal appeals courts with oral arguments scheduled later this week to render decisions "with appropriate dispatch." Two justices -- Ruth Bader Ginsburg and Sonia Sotomayor -- would have denied the Trump administration's request.
The battle over the travel ban dates to the first week of the Trump administration and has involved different countries, restrictions and exemptions.