Union leaders said the move threatened up to 600 jobs and warned of strikes to fight any compulsory redundancies.
Hundreds of BBC staff listened in dismay as Director-General Mark Thompson outlined plans to close 6 Music and the Asian Network despite campaigns by the public and celebrities to keep them open.
Jeremy Dear, general secretary of the National Union of Journalists, said: "In confirming these proposals today, Mark Thompson has put BBC management on a collision course not just with us and the hundreds of BBC staff who face losing their jobs, but with licence fee payers up and down the country.
"Public outrage at the proposed cuts has been overwhelming. A 'Save BBC6 Music' Facebook group has gained almost 90,000 members in just a few days and group members have appealed to us and other BBC unions to organise a joint demonstration.
"We've seen an increase in requests for membership from BBC staff right across the country."
Union leaders will meet Mr Thompson tomorrow and will tell him they will oppose the cuts.
The BBC outlined plans to divert nearly £600 million a year to higher quality content in its drive to become "more disciplined and sharply-focused".
In its wide-ranging strategy review to the BBC Trust published today, titled Putting Quality First, the corporation said the BBC is on course to hit £2 billion in savings by the end of this licence fee period in 2012/13.
From 2013, the BBC plans to spend at least 90p in every licence fee pound on making content and delivering it to audiences.
It wants to halve the number of sections on its website, close "lower performing sites" and spend 25% less.
The closure of teen offerings BBC Switch and Blast! is also recommended.
Spending on imported programmes and films would be cut by 20%, capping it thereafter at no more than 2.5p in every licence fee pound.
Spending on sports rights would be capped at 9p in every licence fee pound.
The report follows accusations the BBC has been crowding out its commercial rivals and venturing into areas it should not be entering.
The corporation has also faced calls for greater transparency and clearer objectives to be set out in order to demonstrate value for money.
Presenter Jarvis Cocker added his opposition to the plans.
He told Radio 4's The World At One: "As far as I can see, a station like Radio 6 is perfectly in the spirit of what the BBC was supposedly set up to do - public service broadcasting. It's airing music that wouldn't get heard otherwise."
Liberal Democrat culture spokesman Don Foster said that while today's report signalled the end of the BBC "roaming wherever it fancied" he feared the stations earmarked for closure were "sacrificial lambs".
Shadow culture secretary Jeremy Hunt said he was pleased the BBC "is taking a long, hard look at exactly what it should be doing, but the truth of the matter is that we need to see actions not words".
Some 33 MPs signed up to a parliamentary motion expressing "deep concern" at the prospect of the two radio stations being closed down.
While it plans to close 6 Music by the end of 2011, the report added that the BBC should maintain its overall investment levels in original radio content aimed at digital services.
"The BBC will also review how some of 6 Music's most distinctive programmes can be successfully transferred to other BBC radio stations, and how its support for new and specialist music can be sustained across the BBC," it said.
The report acknowledged 6 Music reaches 700,000 listeners a week, playing music they do not hear elsewhere.
But the average listener age of 37 "means that it competes head-on for a commercially valuable audience.
"Boosting its reach so that it achieved appropriate value for money would significantly increase its market impact."
The BBC concluded that focusing investment on the "core networks" Radio 1 and 2 was the "most effective and efficient way to deliver popular music on radio".
The report said: "The Asian Network in particular has failed to build on its early growth and has lost around 20% of its listeners over the past three years.
"The service's current scale appears insufficient to justify the level of BBC investment as a network radio service."
Mr Thompson said: "I don't want to pretend that these are easy decisions.
"It's very interesting that politicians say 'Why don't you cut these services?'
"When we start doing that they say 'Have you gone mad?'
"These are difficult decisions and it's painful for the people who listen and watch a given service and for those people that broadcast it."
1Xtra and Radio 7 will be more closely aligned with their parent stations. Over the next 18 months 1Xtra will strengthen links with Radio 1, while Radio 7 will have closer ties with Radio 4 and be re-branded Radio 4 Extra.
BBC Trust chairman Sir Michael Lyons welcomed the report.
He said: "We are clear it heads towards a more disciplined and sharply-focused BBC.
"That will mean some difficult choices. But we will not shrink from those choices where they are in the interests of licence fee-payers."
The BBC said it would recognise the role on television Channel 4 and other broadcasters are likely to play in meeting a gap in public service broadcasting for younger teenagers.
By closing teen-focused BBC Switch, the BBC plans to concentrate on serving these audiences through mainstream services.
The report said that in the main, buying in shows from overseas "cannot be an editorial priority for the BBC".
There will be no acquired series during peak time on BBC1, it said.
The proposals will be subject to a 12-week consultation and Sir Michael promised that the Trust would listen to the public.
He said: "If we find that ... there's massive public concern that we need to take account of then we will go back to the Director-General to rethink the strategy before it's approved."
The BBC plans to simplify its structure, with fewer management layers and "job titles which pass the tests of clarity and common-sense".
The corporation has come under fire for not releasing breakdowns of the individual stars' salaries.
The report said: "The BBC will also be more transparent through the publishing of the total talent costs incurred on an annual basis, allowing licence fee-payers to see whether costs are being controlled but balancing this against the likelihood that full disclosure would lead to fresh upward pressure on pay."
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