The rocket came hours after George Bush, the US president, landed in Israel to celebrate the creation of the state with the Israeli government.
David Chater, Al Jazeera's correspondent in Israel, said: "This is not the ordinary Qassam rocket that is usually fired. They are essentially homemade rockets with not much explosive force.
"This appears to have been a Grad rocket which has a much more explosive warhead on it. It has made a direct hit on a shopping mall in Ashkelon.
"The building has been devastated. We understand there are people trapped inside.
"There are many injuries, including a young baby who is in a critical condition.
"Clearly this was timed to coincide with the visit of President Bush."
Palestinian factions frequently fire homemade rockets across the boundary, although injuries and deaths are rare.
Two Palestinian civilians and three fighters had been killed in the Gaza Strip earlier in the day in an Israeli military raids, Palestinian medical officials have said.
Israeli troops pushed into southern Gaza to target rocket-launching squads at dawn on Wednesday, the Israeli army said.
Two Palestinian fighters were also killed in air raids near the town of Khan Younis, Palestinian doctors said.
Hours later, Israeli tanks raided an area northeast of Gaza City near the border with Israel, killing a 17-year-old youth riding his bicycle and another civilian, doctors said.
Another Palestinian fighter was also killed in the raid.
Al Jazeera's Wael al-Dahduh said that the Israeli incursion into east Khan Younis and Rafah in the southern Gaza Strip was on a wide scale.
"The Israeli military vehicles are still deployed in this area where clashes have erupted between Palestinian resistance fighters and Israeli soldiers who took roof top positions on a number of houses, turning them into military barracks," al-Dahduh said.
"Israeli bulldozers, backed by tanks and helicopters, have also started demolishing lands in the raided area," he said.
The violence came hours before George Bush, the US president, arrived in Israel for a three-day visit.
The fighting has clouded Egyptian efforts to mediate a ceasefire between Israel and Gaza's Hamas rulers.
Hamas said on Wednesday it will go to Egypt to discuss a Gaza Strip truce.
Ayman Taha, a spokesman for Gaza's Hamas rulers, said that Omar Sulieman, the Egyptian mediator, had invited Hamas officials to hear Israel's response to a proposed Egyptian truce deal on Sunday.
Suleiman, who has been trying for months to clinch a truce between Israel and Gaza's leaders, was in Israel this week to discuss the truce efforts.
Hamas and Israel have each put forward tough conditions for a truce to occur.