BBC station Radio 7 is being given a new name - Radio 4 Extra - in an attempt to attract more listeners.
Radio 7 was launched in 2003 but in 2009/10 fewer than one in five British adults were aware of the station, which has an annual budget of £5.8 million.
The BBC Trust, the Corporation's governing body, has now endorsed a management proposal to reposition the speech-based comedy and drama station, which has an audience of 925,000, as Radio 4 Extra.
The BBC also wants to build awareness of Radio 7 by fostering closer editorial ties with Radio 4, which has around 10 million listeners, and by doubling the levels of cross-promotion.
Despite lower awareness of Radio 7 than any other BBC network station, it had been successful, the Trust said.
But it regretted that the BBC's attempt to serve children through dedicated programming on Radio 7 had been unsuccessful.
As a result, it has approved plans to reduce the hours of children's programming from 1,400 to 350 hours on Radio 7 and refocus it on "family-friendly content", saying this would result in a "stronger speech service for children".
The Trust reviews each of the BBC's services at least once every five years.
Today it also published its findings on Radio 4, saying many listeners considered the station a "national treasure".
It endorsed proposals to increase levels of listening among audiences outside of London and the South East and those from ethnic minority groups.
It also said that Radio 4 should address concerns about the imbalance of its international coverage in favour of the US, compared with Europe and the rest of the world.
It endorsed plans to find ways to build loyalty among "younger, lighter listeners", saying that there were "concerns about long-term decline in reach to the 'replenisher' audience group".
Radio 4's £91.3 million budget is nearly double that of the next most expensive BBC radio station, but the Trust said it was "cost-effective" due to its strong performance.
Radio 3, which broadcasts mainly classical music, has a budget of £39.2 million and attracts around two million adult listeners, fewer than any other BBC network radio station and around a third of Classic FM's audience.
It has the highest cost per listener-hour of the main BBC radio stations.
The BBC Trust said there was no compelling evidence to back up concerns that the station's attempt to be more accessible had reduced its quality and distinctiveness.
Radio 3 should become more accessible without alienating its core audience or reducing its distinctive output, it said.
BBC management have raised the possibility of providing an opt-out digital stream at weekends for jazz, world music and special events.
The Trust said it concluded that BBC Radio 3, Radio 4 and Radio 7 are "much loved and highly valued by the millions of licence fee-payers that listen every week".
The report said an hour of Radio 7 content costs £571 to broadcast.
It aims to boost listening to the station by 50% through the changes announced today.
RadioCentre, the industry body for commercial radio, criticised today's report.
Chief executive Andrew Harrison said: "The BBC Trust's call for Radios 3 and 4 to extend their appeal, whilst retaining their distinctiveness, is a contradiction in terms.
"It's like asking a station to become more popular and less popular at the same time. Radio 3 and Radio 4 should be proud of their unique character and do all they can to retain it."
Radio 3 controller Roger Wright welcomed the trust's "recognition of Radio 3's distinctive music and arts output and support for the current direction of the station".
He added: "The trust confirms that Radio 3 represents excellent value for money. We welcome the trust's recommendation that Radio 3 continues to build appeal for potential new listeners whilst, vitally, maintaining the station's unique music, arts and broadcast features of the highest quality."
Radio 4 and Radio 7 controller Gwyneth Williams said: "I warmly welcome the results of the trust's very thorough review of Radio 4. The overwhelmingly positive feedback they received from listeners is a tribute to all the programme makers across the UK on whose creativity and high standards the station depends.
"I am also encouraged by the trust's endorsement of our continuing work to extend the appeal and accessibility of the station while remaining true to our values and our very loyal audience."