Burns takes up his post on the board of governors from the beginning of next month. He replaces Neville-Jones who stood down last December.
Neville-Jones quit the BBC board of governors a year early, after her links with defence firms supplying US forces in Iraq were exposed earlier this month.
Neville-Jones was called upon to stand down after it was revealed she held shares in defence firm Qinetiq. She was reported to hold £50,000 of shares in the firm and last year earned £133,000 from the company as chairman.
Burns said the BBC had an enormous role to play in promoting the UK to audiences overseas and reflecting the world back to viewers and listeners at home.
"I look forward to playing my part as international governor, and to serving licence payers across the UK to ensure the BBC delivers the quality broadcasting they expect," he said.
As well as being British high commissioner to Canada (2000-2003), Burns' career with the Foreign and Commonwealth Office (FCO) also includes time as British ambassador to Israel (1992-1995) and as British consul-general to Hong Kong and Macau (1997-2000).
He was press counsellor and head of British information services in the US at the British embassy in Washington (1983-1986) and head of the FCO news department (1988-1990).
He is also a member of the British-North American committee and a fellow of the Portland Trust. He is a non-executive director of JP Morgan Fleming Chinese Investment Trust and current chairman of the council of Royal Holloway, University of London.
Michael Grade, chairman of the BBC, said: "His distinguished public service career will benefit licence payers throughout the UK and his wealth of experience of foreign affairs and continuing interest in international issues will serve the BBC's international audiences very well."
The appointment is the likely to be the last one made under the current BBC charter which is due to expire in December 2006.
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