Perrigo, which manufactures generic drugs for the over-the-counter market in America, has agreed to pay $6.25 a share in cash plus $10.25 a share in stock for Elan, a premium of about 10.5pc over Elan's closing price on Friday.
Joseph Papa, Perrigo's chief executive and chairman, said the deal would provide a "platform for further international expansion," with the two companies combining to create a "industry-leading global healthcare company".
Perrigo will unlock access to Elan's lucrative royalties from Tysabri, the multiple sclerosis drug, in which it has maintained royalty rights of up to 25pc and generated revenues of more than £1bn last year.
The deal will also enable Perrigo to take advantage of Ireland's 12.5pc corporation tax rate, the lowest in Europe, compared with 35pc in the United States.
"When you put [the deal] together with an Irish domicile that has operational tax synergies, we think it's a really compelling story," Mr Papa told Reuters. Perrigo said the deal would also help the new company make separate tax savings of more than $150m.
The deal ends a bitter takeover saga in which Elan rejected three hostile bids by US investment firm Royalty Pharma amid injunctions, court hearings and a war of words in which Royalty Pharma said Elan was only worth $13 per share because of its "high risk strategy of hastily arranged and value destructive acquisitions".
Perrigo will fund the deal using a $4.35bn bridging loan from Barclays and HSBC, plus cash. The new company will be a namesake of Perrigo, the US company said on Monday, and will be led by its current management team.
Perrigo shareholders are expected to own approximately 71pc of the combined company with Elan shareholders owning the remaining share.
Elan shares jumped by as much as 11pc on Monday morning, to €12.36.