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His appointment sees a UK executive take over the running of Glaxo for the first time since the group was formed by the £114 billion merger of equals between Glaxo Wellcome and SmithKline Beecham in 2000.
Mr Viehbacher, a dual Canadian-German citizen, had been seen as favourite for the role with insiders claiming he enjoyed greater access to Mr Garnier. Mr Stout is an American.
Shares in the group fell 7p to 1323p.
Kevin Wilson, an analyst with Citigroup, said that the announcement, and the implicit relative unanimity of the board, should be taken as a minor positive.
However, others added that Mr Witty’s win could lead to the departure of both Mr Viehbacher and Mr Stout.
Paul Diggle, analyst at Nomura, said: "Glaxo might lose both. I would be very surprised if Mr Stout didn’t move on.
"The best way of keeping Mr Viehbacher may be to let Mr Stout go and then promote Mr Viehbacher.
Mr Garnier, formerly with SmithKline Beecham, took over as group chief executive following the merger seven years ago.
He said that his legacy would be the creation of a new corporate culture at the business — one of "performance with integrity".
He said: “It’s like as a parent, if you educate your children well that’s more important than anything else that you have done specifically for them.”
He added: “I was given an institution that was not in the greatest shape and I made it stronger — that’s all you can say. The chapters of the book don’t matter very much because they are linked to circumstances.”
Glaxo refused to comment on Mr Witty's pay, although he is likely to become one of the best-paid executives in the City.
Mr Garnier took home more than $5.4 million (£2.6 million) last year with a base salary of $1.7 million.
He was famously the target of a shareholder revolt four years ago, when investors rejected a proposed gold-plated package that could have seen Mr Garnier pocket £22 million if dismissed.
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