* Under Egyptian proposal, both sides to hold fire from 0500 GMT
* Israeli envoys due in Cairo to negotiate more enduring deal
* Ground forces quit Gaza, having razed 32 infiltrator tunnels
* Palestinian death toll 1,834; 67 killed on the Israeli side (Adds Israeli military redeployment)
By Nidal al-Mughrabi, Lin Noueihed and Allyn Fisher-Ilan
GAZA/CAIRO/JERUSALEM, Aug 5 (Reuters) - Israel and Islamist Hamas were expected to hold their fire for three days starting on Tuesday, under terms of an Egyptian-brokered truce, while launching negotiations to cement a long-term deal to end a four-week Gaza war.
Israeli ground forces would completely withdraw from the Palestinian enclave in time for the 8 a.m. (0500 GMT) start of the ceasefire, having completed their main mission of destroying cross-border infiltrator tunnels, a military spokesman said.
The spokesman, Lieutenant-Colonel Peter Lerner, said those forces would be "redeployed in defensive positions outside the Gaza Strip and we will maintain those defensive positions" - reflecting Israeli readiness to resume fighting if attacked.
Several previous attempts by Egypt and other regional powers, overseen by the United States and United Nations, failed to calm the worst Israeli-Palestinian fighting in two years.
Gaza officials say the war has killed 1,834 Palestinians, most of them civilians. Israel says 64 of its soldiers and three civilians have been killed since fighting began on July 8, after a surge in cross-border rocket salvoes from Gaza.
Israel was expected to send several delegates to join in negotiations in Cairo to cement a longer term deal during the course of the truce, Israeli officials said.
For now, Strategic Affairs Minister Yuval Steinitz told Israel's Army Radio, "There are no agreements. As we have already said, quiet will be answered with quiet."
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 praised the truce and urged the parties to "respect it completely". Spokeswoman Jen Psaki added that Washington would continue in its efforts to help the sides achieve a "durable, sustainable solution for the long term".
Efforts to cement 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 with whom it eschews any ties.
Besides the truce, Palestinians have demanded a withdrawal of Israeli forces from Gaza, an end to the blockade of the impoverished enclave and the release of prisoners including those arrested in a June crackdown when three teenagers were kidnapped and killed, demands Israel has resisted in the past.
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.
ISRAEL DEMANDS TO DEMILITARISE GAZA
But Netanyahu also wants to disarm Hamas and demilitarise the coastal territory, stripping of their arsenals guerrillas who launched more than 3,300 rockets and mortars at Israel this past month. Hamas has so far ruled that out.
"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, with each side refusing to accept terms put by the other. This time Egypt's initiative could yield results if the parties can convene to negotiate a longer-term deal.
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, it blockades neighbouring Gaza and its current administration opposes Hamas.
Israel had declined earlier in the week to send a delegation to Egypt for truce talks, raising doubts a deal was achievable.
A Palestinian official affiliated with one of the Gaza factions said a temporary ceasefire would help open the door to more comprehensive negotiations.
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.
The violence showed signs of abating early on Tuesday, with few reported incidents. Two rockets were fired at a southern Israeli city, Ashkelon, and Israel's Iron Dome interceptor shot them down. There were no casualties.
On Monday, Palestinians accused Israel of bombing a refugee camp in Gaza City, killing an eight-year-old girl and wounding 29 other people in a disruption of what was supposed to have been a seven-hour humanitarian ceasefire.
A military spokeswoman disputed this, saying Israel had withheld aerial fire for seven hours on Monday, in a move to free up aid supplies and allow some of the hundreds of thousands of Palestinians displaced by the war to go home.
Media in Britain reported that a British aid worker was killed on Sunday during an Israeli strike in Rafah while he was delivering supplies to a hospital. The British Foreign Office said it was looking into the report. (Additional reporting by Eric Beech in Washington, Ori Lewis in Jerusalem, Yasmine Saleh in Cairo; Writing by Allyn Fisher-Ilan; Editing by Ken Wills and Clarence Fernandez)