The school is working with renowned architects Page\Park to finalise the plans and designs for the redevelopment.
The building will provide learning and teaching facilities for Art, Drama, Music and Culinary Arts.
Proposals include developing a main school auditorium, a new dining hall and further administrative and social spaces for pupils and staff.
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A spokesperson for Page\Park, said, "The acquisition by the School of the former Post Office building adjacent to the main school complex has initiated a remarkable vision for an Arts complex second to none in an academic context in the United Kingdom."
Work to 'tidy up' and strip out the building has been completed.
Once the development commences, it will be carried out in several stages over a number of years.
The Rector of the High School of Dundee, Dr John Halliday, said "The ability to think analytically, critically, but also creatively is a crucial component of what we do in school.
"The new centre is a once-in-a-century opportunity for us to ensure that creativity continues to flourish within the education we provide our pupils, for generations to come."
(LM)
