Army spokesman Adolphe Manirakiza said the attacks on Thursday were carried out by the hardline Forces for National Liberation (FNL), the only rebel group still fighting government forces in the small country's 12-year-old civil war.
The FNL was not immediately available for comment.
"The first criminal attack was committed in one church at Muhuta district where a group of FNL fighters fired in the church and killed six people including a pastor. They also injured seven others," Manirakiza said.
He said the rebels shot dead a local government official and his brother and also kidnapped a policeman in the same district.
"We don't know yet why these people were killed, they may be executed because they refused to adhere to the FNL ideology, as the area where they have been killed is an FNL stronghold."
Burundi, a coffee-growing nation of seven million, is slowly emerging from an ethnic conflict pitting majority Hutus against the politically dominant Tutsi minority. Some 300,000 people have been killed since the fighting began in 1993.
Fresh talks between the government and the FNL have raised hopes among diplomats that Burundi can finally lift one of its last big barriers to peace.
Its leader Agathon Rwasa and Burundi President, Domitien Ndayizeye signed on May 15 in Dar es Salaam, Tanzania, a cessation of hostilities deal seen as a significant step toward finally bringing the rebels into the peace process.
But the rebels and the army have clashed sporadically since the truce.
On Tuesday, delegations from Burundi's government and the FNL met again in Dar Es Salaam and agreed to probe violations of the cease fire agreed last month.
- Reuters - Thomson Reuters Foundation
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