Extensive work, including construction of a new roof, will be carried out.
The total cost has yet to be confirmed by the royal household but it will be published next week.
A spokeswoman said there had previously been no running water and the couple had paid for all internal furnishings.
She added asbestos was removed and an overhaul of the electrics was carried out.
Prince William and Kate Middelton’s London apartment was designed by Christopher Wren and was the home of Princess Margaret and Lord Snowdon.
The princess remained there after their divorce and lived there until her death in 2002.
The living space was last refurbished in 1963.
A spokeswoman for the couple said: “This is the duke and duchess’s one and only official residence. It is here that they plan to stay for many, many years to come.
“We also had to take into account the fact that Kensington Palace is a scheduled ancient monument, and all elements of the refurbishment had to be agreed with English Heritage.
“The Duke and Duchess of Cambridge paid privately for all the internal furnishings, including carpets and curtains.
“They were also at pains to ensure that the specification is not extravagant.”
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