They voted by more than 99 per cent to allow the bank to join the Government's asset protection scheme.
The Treasury-backed scheme will act as a large insurance policy for the bank's £282 billion in bad debts.
The Treasury is pumping an extra £25.5 billion into the bank under the proposals - with a further £8 billion ready if needed - taking its stake to 84 per cent.
The vote followed a meeting in Edinburgh in which chairman Philip Hampton said the measures were designed to equip RBS to deal with the "headwinds" it faces over the coming years, but also with the "risks that events might turn out significantly worse than we currently forecast".
Sir Philip warned that there were no alternative capital measures available to the ailing bank, which lost a UK record £24.1 billion last year.
"Bluntly, without these measures RBS would be at risk of full nationalisation, and in that event independent shareholders could very well lose most or all of the value in their shares," he said.
The meeting came a day after the latest bail-out proposals were approved by European competition officials in return for major sell-offs by the bank.
The European Commission has demanded the sale of RBS's Churchill and Direct Line insurance arm and parts of its investment banking business under a wide-ranging restructuring programme in return for the extensive state support.
RBS is also selling 318 branches of the former Williams & Glyn's outlets in England and Wales and its NatWest branches in Scotland.
These represent 14 per cent of its UK network and will cut its retail market share by 2 per cent. Its share of the small business banking market will fall by 5 per cent.
Sir Philip told the general meeting at the bank's Gogarburn headquarters today: "It is possible that we may look back in years to come and, if the severe outcomes against which we have obtained protection have not come to pass, feel that the price was too great.
"Today, however, it is the board's view that the costs of this transaction are commensurate with the substantial risk protection provided to your group."
He also told shareholders: "Since February, the economic environment has become less harsh, and the probability of the downturn continuing much longer now appears lower than it seemed then.
"But these measures are designed to equip RBS to deal not just with the headwinds we know we face over the coming years, but also with the risks that events might turn out significantly worse than we currently forecast - if, for example, economic conditions were to turn sharply downwards again."
During his speech, the chairman also denied that the board had threatened to resign in a row with the Government over bonuses.
The claim emerged after reports that the bank's board had received legal advice they could have to step down en masse if the Treasury vetoed the payouts planned for its top earners.
Sir Philip told the meeting: "Contrary to a number of media reports, there have been no threatened mass resignations of the board, at any time.
"The board are wholly committed to their legal and other duties and will remain so, on the basis that they are able to discharge them."
The board was expected to face tough questions from shareholders on bonuses, but instead they faced a handful on the size of banks, share prices and dividend payments.
Graham Aitken, a retired former employee of RBS, spoke out about the "forgotten victims of the collapse of Royal Bank" - small private shareholders.
Mr Aitken, of West Lothian, said they have "endured the horror of watching their savings basically completely evaporate as a direct consequence of the meltdown of the Royal Bank share price".
He added that the situation was capped by the freezing of share dividend payments and questioned when they were likely to restart.
Sir Philip said the loss shareholders had suffered in recent years was "extremely regrettable" but he insisted it was too soon to say when dividend payments could be restored.
Felix D'mello, from Edinburgh, raised concerns about the bank losing some of its "flagship businesses".
He said: "I'm just a little concerned that, having done all the asset-stripping, there won't be enough critical mass left to be able to pay us a dividend."
Another shareholder asked whether the board had looked at focusing solely on retail banking, distancing themselves from what he called the "Monte Carlo people".
He told the meeting there was a lot of comment in the financial press, saying that the banks really are too big and should be shrunk in size so that if they do fail again they do not bring the economy "down on its knees as they have done".
Sir Philip said the issue of bank size and the relationship between retail and investment banking was part of an ongoing regulatory debate and it was too soon to come up with a definitive conclusion about the shape of the bank in years to come.
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