WASHINGTON — The Trump administration moved aggressively on Thursday to fulfill one of the president’s most contentious campaign promises, banning entry into the United States by refugees from around the world and prohibiting most visitors from six predominantly Muslim countries.

Freed by the Supreme Court to partly revive President Trump’s travel ban, administration officials said the American border would be shut to those groups unless specific individuals can prove they have close family members living in the United States, or are coming to attend a university or accept a job offer.

Officials said those exceptions would be defined narrowly. In a lengthy cable sent to embassies and consulates around the world, officials said that extended family connections would not be enough to evade the president’s ban on entry. Parents, including in-laws, are considered “close family,” but grandparents are not, for instance. Stepsiblings and half-siblings will be allowed, but not nieces or nephews.

Late Thursday, the Trump administration added people who are engaged to be married — who originally were not considered to be close family members — to the list of sufficient connections.