The total cost of the sale has not been disclosed but BBC Scotland News understands it was below market value.
RSPB Scotland said acquiring the islands came at a crucial time, with more than 70% of seabirds facing decline.
It said threats come from climate change, food scarcity, invasive species and human activity.
Bass Rock has been a Site of Special Scientific Interest (SSSI) for almost 75 years and is home to about 100,000 gannets.
Anne McCall, director of RSPB Scotland, said the organisation was “thrilled” to be taking ownership of the islands and would be working with the Scottish Seabird Centre to protect and enhance the bird colonies.
“With the pressures facing Scotland’s seabirds intensifying, we have a responsibility to act,” she said.
“This partnership gives us the opportunity to drive forward seabird recovery at this crucial time.”