The 26,000-square-foot building on Liberty Island will provide information and exhibits for visitors, after access to the current museum inside the statue was restricted following the 9/11 terrorist attacks.
It will be located behind the iconic copper statue, designed by Gustave Eiffel – the French engineer behind the eponymous Parisian tower – and completed in 1886.
FXFOWLE's design includes a wide external staircase that will lead to a rooftop viewing platform, providing views of both the statue and the Manhattan skyline beyond.
Grass will cover the remainder of the roof, including portions that will slope down to a circular plaza between the museum and the landmark.
"From the start, the design of the Statue of Liberty Museum was conceived as an extension of the park," said FXFOWLE partner Nicholas Garrison. "The goal was to engage with the park's formal, axial plan and respond to its spectacular setting."
The new $70 million (£56.3 million) structure will feature "bird-safe" glass walls and portions of bronze cladding. It will be raised above 500-year flood levels and built to withstand hurricane force winds.
"The island's landscape is lifted and merged with the architecture to create space for the Museum in a new geology," Garrison said. "The building's angular forms and spaces are shaped by its views and the irregularity of the water's edge, celebrating liberty."
Inside, exhibits created by ESI Design will include contents and artefacts relating to the history of the statue – a gift from France.
"With state-of-the-art exhibitions and iconic artefacts including the statue's original torch, the new Statue of Liberty Museum will ensure that future generations know, understand, and appreciate all that Lady Liberty represents in America and around the world," said John Piltzecker, Statue of Liberty National Monument and Ellis Island superintendent.
The Statue of Liberty-Ellis Island Foundation, which restored the statue for its centennial, has named fashion designer Diane von Furstenberg as chair of the campaign to raise $100 million (£80.5 million) for the museum's construction and upkeep.
Von Furstenberg and husband Barry Diller are also benefactors for the Thomas Heatherwick-designed Pier 55 park under construction over the Hudson River.
"Lady Liberty is the symbol of everything America is about: freedom, hope, possibility and resilience," said von Furstenburg. "It is she that millions of immigrants saw first as they arrived in this country, their hearts full of dreams for a brighter future."
"Now it is my hope that the Statue of Liberty and her incredible story will live on and on, inspiring generations for years to come," she added.
Access to the museum, due to open in 2019, will be included in the price of ferry tickets to visit Liberty Island and Ellis Island. Liberty Island will remain open to visitors during the construction period.