The Sri Lankan government immediately hailed the visit as a sign of confidence in the island's security arrangements.
"The Bangladeshi government has accepted our security guarantees making the tour possible," Sri Lankan Sports Minister Gamini Lokuge told AFP, adding that Colombo would provide security for the team during its five-week stay.
"This shows that Sri Lanka is safe for anyone to travel," he said.
Bangladesh's government has warned its nationals not to travel via the troubled island, and cricket officials said the authorities in Dhaka would have the final say on whether the tour could go ahead.
The Liberation Tigers of Tamil Eelam (LTTE) have carried out two separate air raids in the Sri Lankan capital since the end of March, the first of which targeted an air force base adjoining the island's only international airport.
At least nine people have died and more than 40 civilians were injured in two roadside bombs targeting troops in the city and its suburbs since then.
South Africa abandoned their tour of Sri Lanka in August last year after a bombing in Colombo blamed on the Tamil Tigers, which targeted the Pakistani ambassador.
The three-match Test series begins on June 25, while the three-match ODI series starts on July 20.