Hamas spokesman Sami Abu Zuhri said the Islamist group had also informed Egypt "of its acceptance of a 72-hour period of calm," beginning on Tuesday.
The U.S. State Department welcomed the truce and urged the parties to "respect it completely". Spokeswoman Jen Psaki added that Washington would continue its efforts to help the sides achieve a "durable, sustainable solution for the long term".
Efforts to turn the ceasefire into a lasting truce could prove difficult, with the sides far apart on key demands, and each rejecting the other's legitimacy. Hamas rejects Israel's existence, and vows to destroy it, while Israel denounces Hamas as a terrorist group and eschews any ties.
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Besides the truce, Palestinians demand an end to the Israeli-Egyptian blockade on impoverished Gaza and the release of prisoners including those
Israel arrested in a June crackdown in the occupied West Bank after three Jewish seminary students were kidnapped and killed.
Israel has resisted those demands in the past.
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Palestinian Foreign Minister Riad al-Malki planned to visit the International Criminal Court in the Netherlands on Tuesday as he pushes for a war crimes case against Israel. Both sides have traded allegations of war crimes during the Gaza assault, while defending their own actions as consistent with the international law.
Israel: Demilitarize Gaza
Lerner said the army overnight destroyed the last of 32 tunnels located inside Gaza and which had been dug by Hamas for cross-border attacks at an estimated cost of $100 million.
"Today we completed the removal of this threat," he said.
Israeli officials say, however, that some tunnels may have gone undetected and that the armed forces are poised to strike at these in the future.
Prime Minister Benjamin Netanyahu also wants to disarm Hamas and demilitarise Gaza, after guerrillas launched more than 3,300 rockets and mortar bombs at Israel this past month. Hamas has ruled that out.
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"For Israel the most important issue is the issue of demilitarisation. We must prevent Hamas from rearming, we must demilitarise the Gaza Strip," Netanyahu spokesman Mark Regev told Reuters television.
Since the fighting began, several previous truces barely held. Regev said Israel had accepted Egypt's terms weeks before Hamas, and expressed a wish that the truce would last: "I hope this time we see the ceasefire work that's good for everybody."
Egypt has positioned itself as a mediator in successive Gaza conflicts but, like Israel, its current administration views Hamas as a security threat.
Besides the loss of life, the war has cost both sides economically. Gaza faces a massive $6-billion price tag to rebuild devastated infrastructure. Israel has lost hundreds of millions of dollars in tourism, other industry, and fears cuts in overall economic growth this year as well.