At least three local residents were wounded in the early morning attack by an F-16 jet that destroyed what was left of the eight-story building.
The ministry, which was empty at the time of the strike, had already been badly damaged in a raid on Thursday.
Israel said the ministry was targeted because it is run by a senior figure in the Hamas movement, Foreign Minister Mahmoud Zahhar.
Israel has bombed several buildings used by Hamas and the Palestinian government, including the office of Prime Minister Ismail Haniyeh.
The Israeli military confirmed the attack, saying it was “part of ongoing operations”.
At least 50 apartments and four ministries in the same compound were also extensively damaged in the explosion.
Israel launched an offensive in the Gaza Strip on June 28 to try to retrieve a captured soldier and halt Palestinian rocket fire.
It widened that assault on Sunday, killing a Palestinian civilian in southern Gaza and five militants in the north.
Israeli forces moved back into northern Gaza early yesterday, 10 days after completing a two-day sweep aimed at confronting militants who have been firing rockets at Israel.
Troops and tanks, backed by helicopters, moved towards the town of Beit Hanoun, an area often used by militants for launching rockets into Israel.
At least three Palestinians were killed and eight wounded by Israeli air raids, Palestinian medical sources said.
Israel says its Gaza offensive was meant to win the soldier’s release and to end Palestinian rocket fire at Israeli communities around Gaza.
Yuval Diskin, the head of Israel’s Shin Bet security service, reportedly told Israel’s Cabinet he believed the kidnapped soldier was still alive.