The 90-year-old monarch will remain in residence during the works, which are set to begin in April 2017.
The Queen's household budget has been awarded a 66 per cent pay rise to pay for the 10-year refit of the palace in central London, designed to future-proof the ageing building.
"Buckingham Palace is one of the most iconic buildings in the world, and this programme is designed to extend its working life by a further 50 years. On completion of the work, we'll have a palace fit for purpose until 2067," said Tony Johnstone-Burt, master of the Queen's household.
"We take the responsibility that comes with receiving these public funds extremely seriously indeed. Equally, we are convinced that by making this investment in Buckingham Palace now we can avert a much more costly and potentially catastrophic building failure in the years to come," Johnstone-Burt said.
The Treasury said the UK Parliament would hold the royal household to account throughout the process to ensure maximum value for taxpayers' money.
"These urgent works have been properly costed and will ensure the palace can continue its centuries-long tradition of being the working house of our monarch. We will ensure every penny spent achieves the greatest value for money," Gauke said.
Republic, which campaigns for the abolition of the monarchy, called for an "independent inquiry and full disclosure" into the use of taxpayers' money.
As the British monarch's administrative headquarters, the Queen spends a third of the year hosting events at Buckingham Palace.
When in town, she holds weekly audiences with the prime minister, and every year welcomes more than 50,000 people as guests to state banquets, dinners, receptions and garden parties.
The palace has 775 rooms, including 19 state rooms, 52 royal and guest bedrooms, 188 staff bedrooms, 92 offices and 78 bathrooms.
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