"THE CEASEFIRE IS NOW IN EFFECT. PLEASE DO NOT VIOLATE IT!" Trump said in a Truth Social post.
When Trump announced on Monday what he called a complete ceasefire to end a 12-day war, he appeared to suggest that Israel and Iran would have time to complete missions that were underway, at which point the ceasefire would begin in a staged process.
Witnesses said they heard explosions near Tel Aviv and Beersheba in southern Israel before Trump's statement.
Israel's military said six waves of missiles were launched by Iran and Israel's national ambulance service said four people were killed in Beersheba, the first reported deaths in Israel since Trump announced the ceasefire.
The official, providing details of the ceasefire on condition of anonymity, said Israel agreed to it so long as Iran does not launch fresh attacks. Iran signaled that no further attacks would take place, the official said.
Iran's semi-official SNN news agency reported on Tuesday that Tehran fired its last round of missiles before the ceasefire came into effect.
A senior White House official said Trump had brokered a ceasefire deal in a call with Israeli Prime Minister Benjamin Netanyahu and Israel had agreed so long as Iran did not launch further attacks.
"On the assumption that everything works as it should, which it will, I would like to congratulate both Countries, Israel and Iran, on having the Stamina, Courage, and Intelligence to end, what should be called, 'THE 12 DAY WAR'," Trump wrote on his Truth Social site.
An Iranian official earlier confirmed that Tehran had agreed to a ceasefire, but the country's foreign minister said there would be no cessation of hostilities unless Israel stopped its attacks.
Abbas Araqchi said early on Tuesday that if Israel stopped its "illegal aggression" against the Iranian people no later than 4 am Tehran time (0030 GMT) on Tuesday, Iran had no intention of continuing its response afterwards.