Rescue workers are still working at the scene of the blast, which occurred in the Kiryat Menahem sector during the morning rush hour.
The bus, travelling towards the centre of the city, was full when the passenger blew himself up at 0715 local time (0515 GMT). Many of the others on board were children, officials said.
A spokesman for Prime Minister Ariel Sharon said Israel would "respond rapidly to stem a wave of terrorist attacks".
Spokesman Raanan Gissin told the French news agency, AFP, that Mr Sharon had met the defence minister and "adequate orders had been given to the army".
The military wing of the militant Hamas movement, Izz el-Deen al-Qassam Brigades, has claimed responsibility for the attack and named the suicide bomber as Na'il Azmi Abu al-Hal, Hezbollah's al-Manar television station reported, according to Reuters.
The attack on Thursday was the first fatal bombing in Jerusalem since 31 July, when seven people were killed at the Hebrew University. It is the deadliest since June 18, when 19 civilians died in a suicide attack.
'Bloodshed agenda'
According to local media, one of the victims is a 13-year-old girl.
Local resident Ariel Gino told Reuters news agency the explosion was so loud he thought his roof had come off.
"I rushed out and saw people lying on the street. Some were screaming, some were crying. There were about five or six people still in the bus. They weren't moving. Some on the street had blood on their faces, others had burns," he said.
An Israeli radio reporter said schoolbooks lay scattered by the charred remains of the bus.
Government spokesman David Baker condemned the attack.
"Palestinian terrorists continue to cling to the bloodshed agenda and they are satisfied only when innocent Israeli lives are taken," he said.
BBC Jerusalem correspondent Jeremy Cooke says Mr Sharon, will be under intense pressure to take a hard line against Palestinian extremists as he seeks re-election in the January poll.
Militant call
Police believe the suicide bomber entered Jerusalem from Bethlehem.
A senior official of the Islamic militant group Hamas said there was no reason for suicide attacks to be stopped.
"Such operations must go on," Abdul Aziz Rantissi told the Arabic TV station al-Jazeera, adding that the "vast majority" of Palestinians supported them.
Palestinian militants have been carrying out a campaign of suicide bombings and other attacks on Israeli targets during the two-year-old uprising.