An Associated Press reporter on the scene early on Monday saw clouds of smoke rising from the already damaged building after the strike. At least five people were injured, rescue workers said. The Israeli military refused to comment.
The blast collapsed an eight-storey wing of the building and damaged houses in a wide area around it.
The wounded were taken to nearby houses. The ministry was thought to be empty at the time of the strike.
Israel had bombed the same building on Thursday, saying it was used to plan terror attacks. The Palestinian foreign minister is Mahmoud Zahar, a Hamas leader.
Another missile was aimed at a Hamas office in the Jebaliya refugee camp, setting it on fire, residents said. No one was hurt. Also, an aircraft fired a missile at a farm building where fighters were thought to be hiding.