Created by designers Franc Fernandez and Nichola Formichetti, the controversial hand-sewn garment will feature as part of the museum's 'Women Who Rock: Vision, Passion, Power' exhibition.
The two-floor interactive exhibition features artefacts, videos and listening stations covering more than 70 artists, and also allows visitors to film a short story or moment of inspiration related to women in rock.
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With the singer's decision to wear the raw-meat dress receiving both criticism and praise, Gaga previously explained she wore the gown to prove she "is not a piece of meat".
She said: "Well, it is certainly no disrespect to anyone that is vegan or vegetarian. As you know, I am the most judgment-free human being on the earth. However, it has many interpretations for me this evening.
"If we don't stand up for what we believe in and if we don't fight for our rights, pretty soon we're going to have as much rights as the meat on our own bones. And, I am not a piece of meat."
However, celebrity hair stylist Sam McKnight – who created the platinum wig Lady Gaga wore to the Video Music Awards – wasn't so impressed with the creation, and said it was "very smelly" and attracted flies.
He explained: "It was smelly and very difficult to place a steak on top of a wig. There were a few flies."