BAM will retain a large proportion of the frame and envelope of the former factory while creating the new UTC Royal Borough of Greenwich.
The contract involves refurbishment of more than 2,700 sq metres as well as building 2,600 sq metres of new space.
UTC Royal Borough of Greenwich will specialise in engineering and construction courses for 600 pupils aged 14 to 19.
The building will be ready by September 2013.
The college is one of 24 UTCs announced in last year’s Budget and is BAM’s second win on the programme after picking-up the first UTC in Hackney late last year.
Regional Design Manager, Mike O’Donohoe said: “It is very pleasing to secure more work on the UTC programme, which is central to the government’s education aspirations.
“This award shows we understand the specific ambitions of the UTC programme and in particular how our designs respond to the specialisms.
“Converting a former factory into a state-of-the-art education facility is a challenging job and we look forward to using our skills and experience to deliver a first class facility for the local community.”