The SANAA-designed National Gallery of Hungary is one of five new museums to be built in the park as part of regeneration project called Liget Budapest. It will stand on the site of the Petőfi Events Hall and is expected to open in summer 2019.
The design features concave roof planes and slanted terraces that extend into the parkland.
SANAA and Snøhetta were initially awarded joint first place in a restricted competition to design the Hungarian gallery in April this year, but in a second round of technical and financial negotiations SANAA was awarded the contract.
"Ours is a contemporary form of public space – one where city, gardens and exhibition rooms become part of an experiential whole," said SANAA co-founders Kazuyo Sejima and Ryue Nishizawa in April. "It is a museum that fluctuates with seasonal shifts."
The Pritzker award-winning studio has designed several other museums including an art gallery in Sydney and the Louvre Lens. SANAA has also recently completed a sinuous walkway of glass, concrete, steel and wood that winds across the landscape of a nature reserve in Connecticut.
An architectural competition to re-landscape the area around the museum will launch in December 2015, helping to extend the park's green areas by five per cent.
"The built-up area of the City Park will remain under seven per cent, as prescribed by law, while tens of thousands of square metres will be added to the park's green surface, thus extending it from the current 60 per cent to 65 per cent," said a statement from Városliget Zrt, who is heading up the project.
"The dialogue between the park and the built environment is a priority in the implementation of the Liget Budapest Project, aimed at the renewal of the City Park."
The New National Gallery of Hungary will sit alongside the Sou Fujimoto-designed House of Hungarian Music.