Israel is to launch a wider military offensive in Gaza, after a homemade Palestinian rocket hit a major Israeli city for the first time.
Israeli planes bombed the interior ministry in Gaza overnight
A meeting of Israel's security cabinet instructed the army to begin a long battle focusing on "institutions and infrastructure facilitating terrorism".
"The rules of the game in dealing with the Palestinian Authority and Hamas must be changed," a statement said.
The attack on Ashkelon caused little damage and no casualties.
The scope of the Israeli response remains unclear, but is likely to involve more shelling of northern Gaza, reports say.
Meanwhile, reports from northern Gaza say there have been clashes between Israeli troops and Hamas militants.
Israel has launched more than a week of military raids on Gaza in response to the capture of one of its soldiers by Palestinian militants.
The Israeli cabinet meeting said the aim of the military activity was "damaging" Hamas in the West Bank and Gaza, and "limiting movement of terrorists by severing the Gaza Strip and targeting terror infrastructure".
Reports say it means troops will be sent deeper into Gaza and step up attacks on members of the Hamas movement.
Gaza's two northernmost towns, Beit Hanoun and Beit Lahiya, will be surrounded by troops, and a no-go area imposed by aircraft and artillery will be enlarged, reports say.
Many residents of the two towns are reported to be preparing to leave.
"The Israelis want to hit resistance fighters but they are hiding behind houses, then the Israelis open fire anyway," said one resident of Beit Hanoun.
Dozens of Israeli armoured vehicles have edged into Gaza since the capture of Cpl Gilad Shalit in a cross-border raid by militants on 25 June.
A larger force is poised to enter in the north, from where the crudely-manufactured Qassam rockets are fired, and the south, where the 19-year-old tank gunner is thought to be hidden.
Israeli troops pulled out of the 45km-long territory, home to 1.4 million Palestinians, last summer after an occupation lasting 38 years.
Gaza school bombed
Overnight, Israel carried out more air raids in Gaza, including the second missile attack against the Palestinian interior ministry complex in a week. At least four people were injured.
SOLDIER CAPTURE TIMELINE
Sun 25 June: Cpl Shalit captured in cross-border attack
Mon 26 June: Palestinian captors propose prisoner exchange
Weds 28 June: Israel enters southern Gaza after air strikes
Thurs 29 June: Israel detains dozens of Hamas officials
Sat 1 July: Captors demand 1,000 prisoners be released
Mon 3 July: Captors issue deadline of 0300 4 July
Tues 4 July: Israel: "Shalit still alive". Rocket hits Ashkelon
Other attacks hit a suspected Hamas military training ground and a school run by Hamas.
Palestinian medics say Israeli troops on Gaza's northern border opened fire at suspected looters in an industrial zone, wounding a 13-year-old boy in the head.
Later, at least one Palestinian was killed in an explosion that ripped through a house in eastern Gaza City. Israel denied involvement in what appeared to have been an accident while the men were handling explosives.
Despite the expiry of a Tuesday deadline set by the militants for Israel to free Palestinian prisoners, Israel said the Franco-Israeli Cpl Shalit was still alive.
Meanwhile, five Hamas politicians detained by Israel last week, have appeared in an Israeli military court on the outskirts of Ramallah, in the West Bank.
The men, four members of parliament and a cabinet minister, were among more than 60 Hamas politicians detained last Thursday, following the capture of Cpl Shalit.
In a separate development, Israeli troops killed veteran Palestinian militant, Mahmoud Haimoor, during a raid in the West Bank town of Jericho.