Friday's violence comes after almost four months of protests by Palestinians along the fence with Israel.
Earlier, the U-N called on Hamas and Israel to scale down the violence.
Since the current round of demonstrations began on March 30, more than 140 Palestinian protesters have been martyred - and thousands more injured - by Israeli army gunfire.
At least four Palestinians have been killed by Israeli fire in a fresh round of attacks on the besieged enclave.
"Hamas took a huge blow yesterday and through the Egyptians asked for a ceasefire", said a senior Israeli official, who spoke on condition of anonymity.
Israel's latest bombardment of Gaza comes just days after Israel further intensified its collective punishment of the Palestinian people by imposing "genocidal" trade restrictions on the occupied territory.
The soldier was the first member of Israel's army to be killed on the Gaza front since a 2014 war between Israel and Hamas, a military spokesperson said.
On Friday, Palestinian gunmen killed an Israeli soldier and the Israeli military launched dozens of strikes that killed three Hamas fighters.
"Hamas is forcing us into a situation where we will have no choice but to embark on a large and painful military operation", Lieberman was quoted as saying by Israeli Hebrew-language daily Yedioth Ahronoth.
Surprise weakness of inflation data stirs debate over interest-rate rise
House prices also increased the slowest in five years according to the ONS, with flat prices staying the same for a year. That was largely down to the rising price of fuel, with petrol prices hitting their highest in four years in the month .
"With Egyptian and United Nations efforts it has been agreed to return to the era of calm between (Israel) and Palestinian factions", he said, as cited by Reuters.
A ceasefire announced by Hamas largely held on Saturday after a wave of deadly air strikes across the Palestinian enclave sparked by the death of an Israeli soldier shot near the border.
After dark on Friday, the Israeli military said that Gaza militants had launched three rockets into Israel, of which two were intercepted by its Iron Dome defence system. Israel has carried out such strikes on Gazans launching incendiary kites and balloons at southern Israel.
Israel's military said it had targeted "eight military posts" belonging to Hamas.
Israeli media reported Prime Minister Benjamin Netanyahu was receiving an emergency briefing from the army on the situation.
The same day, around 200 rockets and mortars were fired at Israel from Gaza and four Israelis were wounded when a rocket hit a house in the nearby Israeli city of Sderot.
The ceasefire is the third between Israel and Hamas to be brokered by Egypt this year and the second truce to be agreed this week after a day-long fighting last Saturday.
"If Hamas continues to fire rockets then Israel will respond in a much tougher way than they think", Israel's Defense Minister Avigdor Lieberman said in a statement Friday, the BBC reported. He said they would be returned only as part of a deal similar to that in 2011 under which captured Israeli soldier Gilad Shalit was released in exchange for more than 1,000 Palestinians jailed in Israel.