The regulator said the scale of the fine reflects the "extraordinary" nature and seriousness of the BBC's failures and the resulting breaches of the Broadcasting Code.
Two episodes of Brand's show breached the Code, broadcast on BBC Radio 2 on October 18 and 25.
Ofcom said the BBC broadcast explicit, intimate and confidential information about Georgina Baillie, Sachs's granddaughter, in both programmes without their consent.
Ofcom said: "This not only unwarrantably and seriously infringed their privacy but was also gratuitously offensive, humiliating and demeaning."
Ofcom said broadcasters should be allowed to enjoy the creative freedom to explore issues and ideas without undue interference.
The watchdog said: "Creative risk is part of the BBC's public service role, however, so is the management of that risk.
"In this case, Ofcom's investigation revealed that despite the Russell Brand show being considered by the BBC to be 'high risk' prior to these episodes, the broadcaster had ceded responsibility for managing some of that risk to those working for the presenter, Russell Brand.
"The presenter's interests had been given greater priority than the BBC's responsibility to avoid unwarranted infringements of privacy and minimise the risk of harm and offence and to maintain generally accepted standards."
In the furore that followed the calls, Brand resigned and Ross was suspended without pay for three months.
The BBC Trust, the corporation's governing body, said today that it regretted "that these serious breaches by the BBC have led to a financial penalty being applied by Ofcom and the loss of licence fee-payers' money as a result".
The Trust's Editorial Standards Committee (ESC) concluded that the breaches were serious and that the content was so grossly offensive that there was no justification for its broadcast.
The ESC found that the comments included in the broadcast were "an abuse of the privilege given to the BBC to broadcast to its audiences".
It identified three failings: a failure to assert editorial control by Radio 2, a failure to follow compliance systems, and a failure of editorial judgment.
The Trust instructed BBC management to broadcast an on-air apology, strengthen editorial controls around any "high risk" programme, and ensure that existing guidelines are complied with.
A statement from BBC management said: "As we said last October, this material should never have been broadcast and we apologised unreservedly for that.
"We note that Ofcom has found that senior management acted swiftly to mitigate the offence and damage caused by the breaches of the Code.
"The BBC has since taken comprehensive action to deal with what were unacceptable failures in editorial judgment and compliance which led to the broadcast.
"This has included a comprehensive review of compliance across Audio and Music, and a new policy ensuring presenters and their agents cannot be the executive producer responsible for compliance on their own shows.
"In addition and as is well-known, two very senior managers and Russell Brand resigned and Jonathan Ross was suspended without pay for 12 weeks."