JERUSALEM - Israel's top security official warned yesterday that Gaza extremists can hit more Israeli cities with longer-range rockets, on a day when rockets exploded in border towns and a coastal city after an Israel-Hamas truce expired.

Only one Israeli was slightly wounded in the barrage of 19 rockets and three mortar shells by nightfall. But after a weekend of heavy rocket attacks - and two Israeli air strikes in response - Israel's government threatened to strike back hard.

One rocket exploded in Ashkelon, a city of about 120,000 on the Mediterranean coast 10 miles north of Gaza. In the past, Israel has responded harshly to attacks there.

Yuval Diskin, the head of the Shin Bet security service, warned Israel's cabinet that Gaza's Hamas rulers now have rockets that can reach the larger city of Ashdod farther north on the Mediterranean coast and even the outskirts of Beersheba, 30 miles to the east. Such attacks would increase the likelihood of an Israeli invasion of Gaza.

"The scenarios are clear, the plans are clear, the determination is clear, and so are the ramifications of each of the steps," Israeli Prime Minister Ehud Olmert said at his cabinet's weekly meeting. "A responsible government is not happy to go to war, but does not evade it."

The government has been under heavy pressure to react to the rocket fire, but the military has been wary of doing so for fear of casualties. Past large operations have not succeeded in stopping the rockets.

A truce between Israel and Hamas expired Friday after six months. The truce had frayed since early November, and rocket fire at Israeli towns has been increasing in recent days.

"The Hamas government in Gaza must be toppled, the means to do this must be military, economic and diplomatic," said Foreign Minister Tzipi Livni, in the running to become prime minister in elections Feb. 10. "Whenever they shoot at Israel, Israel must respond."

Defense Minister Ehud Barak told the cabinet broad military action is increasingly likely. "In order to return to a calm like six months ago, we will probably need a wide-scale operation," he said, according to a meeting participant who spoke on condition of anonymity under government guidelines.

Israel carried out two air strikes yesterday aimed at rocket launchers in Gaza, the military and Palestinian officials said. No one was hurt.