A second blast occurred outside the gate of a government-owned slaughterhouse in the city but no one was hurt, police said.
The bus bomb, which exploded about 9.45am (0645 GMT) in the capital's outlying Kirkos district, tore apart the rear of the 11-seat bus. The bus owner, Berhanu Gebremichael, told reporters that one person was killed and three others were wounded and hospitalised.
Demsash Hailu, Ethiopia's top police official, confirmed the explosion was caused by a bomb.
Authorities did not say who they thought was behind the bombing.
It was the first death in a wave of attacks that began in January with minor blasts targeting public buildings and hotels.
On 7 March, three explosions wounded four people at a restaurant, a market and outside a school.
The unexplained blasts have increased political tension in Addis Ababa, which was shaken by two bouts of unrest after disputed parliamentary elections in May.
European Union observers said that widespread violations undermined the result of the elections. More than 80 people were killed in subsequent clashes between police and protesters.
Authorities have blamed the bombings on groups ranging from militant elements of the political opposition to southern separatist rebels of the Oromo Liberation Front, and on a Somali militant Islamist group.
However, there have been no claims of responsibility, no concrete evidence and no indication beyond police statements that the explosions were politically motivated.
Ethiopia's government has also claimed that the plastic explosives used in those blasts were smuggled from neighbouring Eritrea and used by what it called Eritrean-backed "terrorists".
Eritrea, which has been in a dispute with Ethiopia over their border since a 1998-2000 war that killed 70,000 people, ridiculed the charges.