The £1.8m lottery award will go towards the restoration project by Dundee Heritage Trust.
The total cost to renovate the derelict A-listed buildings on the site is £2.9m.
The museum celebrates the city's historic important as a textile production hub, and this initiative will double the size of the current facility.
Proposals for the transformation will allow some 971 sq m of new exhibition space, the restoration of a Boulton and Watt bean engine, the creation of a family history research site and an expansion to the museum's education programme by hiring a new learning officer.
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Gill Poulter, Heritage Director for Dundee Heritage Trust, which operates the site, is quoted by the BBC as saying: "The project gives us the opportunity to create something new and different that will have a dynamic future of its own but which is complementary to the existing jute museum, helping to protect and enhance it.
"We are incredibly grateful to the Heritage Lottery Fund for supporting this project from its very beginnings, seeing its transformative potential and being as excited as ourselves about all the opportunities it presents."
Colin McLean, Head of the Heritage Lottery Fund in Scotland, added: "This is an exciting time for Dundee as it develops as a cultural visitor destination so we are delighted that the heritage of the industry on which the town once thrived is playing a key role in that vision.
"Incorporating these last two buildings into the existing five-star visitor attraction will complete the transformation of Verdant Works with a stunning new space for exhibits and museum events."
In addition to the HLF, the project is being funded by £500,00 from Historic Scotland, with added support from Leisure and Culture Dundee.
(JP)
