The Australian contractor was picked for the major mixed-use project by Chinese developer Wanda One in January.
Balfour Beatty pulled out of its pre-construction services agreement with the developer in November after failing to agree terms on the contract.
In April CI-One Construction, a joint venture between Interserve and China State Construction Engineering Corporation, had walked away after the firms could not agree a price.
One Nine Elms, part of a number of luxury developments in the South London area of Battersea, comprises two towers of 42 and 58 storeys, which are due for completion in 2019.
Spanning an area of 1.14 million sq. ft, the development will accommodate 437 homes, a Wanda Vista Hotel, a business centre, as well as 3,584 sq ft of retail space.
The inner-London high-end market has been shaken following the Brexit vote last June and the introduction of a 3% Stamp Duty surcharge in April, with some investors pulling out and projects losing in value.
However, Wanda One said it remained committed to the project and the market.
UK director Tim Hawthorn said: “We are delighted to announce the appointment of Multiplex as our main contractor for the delivery of One Nine Elms. Following the EU Referendum, this important announcement further demonstrates our commitment to deliver an iconic mixed-use development in the heart of London.”