The firm has been chosen to redevelop the East Wing of Alexandra Palace in the north of the capital.
The £27 million project, designed by architects Feilden Clegg Bradley Studios, includes protecting the East Wing's historic features, stripping out dilapidated modern fittings and asbestos removal.
The contract award marks the culmination of years of planning and months of negotiations to secure funding from the Heritage Lottery Fund (HLF) and Haringey Council.
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The HLF will contribute nearly £19m towards the scheme, with the council providing nearly £7m.
Overall, the project will restore the former BBC studios, bring the Victorian Theatre back into use as a performance venue and redevelop the East Court entrance hall.
Work is scheduled to be complete by 2018.
Chris Tredget, MD of Willmott in north London, said: "Many of our team grew up enjoying the delights of Alexandra Palace, so we're delighted to have a lead role in shaping its exciting future as North London's most iconic building.
"We have extensive experience of working with listed buildings and we'll carry out the refurbishment with full consideration of the community around Alexandra Palace and Park."
(LM/MH)
