Sturgeon, who has been First Minister of Scotland since 2014, told the United Kingdom prime minister that her mandate for another plebiscite was "beyond question" and argued that it would be "democratically indefensible" if May failed to grant the vote.
The Scottish Parliament this week voted in favour of Holyrood having the power to hold a referendum.
On Tuesday, Scotland's devolved parliament voted to hold a referendum on secession in 2018 or 2019, but the British government in Westminster must give its approval before any such poll can be held.
LONDON (AP) Scotland's first minister formally requested a second referendum on independence on Friday, declaring that Scots have the right to exercise their right of self-determination.
The Survation survey, conducted after the Prime Minister said she will refuse the request for a referendum, found most Scots also thought Westminster should not be allowed to block a new vote.
The Scottish government posteda picture of Sturgeon on its Twitter page with a caption that read, "First Minister Nicola Sturgeon in Bute House, Edinburgh, working on final draft of Section 30 letter to Prime Minister Theresa May".
"I am. writing to begin early discussions between our governments to agree an Order under section 30 of the Scotland Act 1998 that would enable a referendum to be legislated for by the Scottish Parliament", Sturgeon wrote in the letter.
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About 62% of Scottish voters backed Remain in the European Union referendum and the SNP's manifesto for last year's Holyrood elections said another ballot on independence should take place if there were a "material change in circumstances" from the previous ballot in 2014 such as Scotland being removed from the European Union against its wishes.
David Mundell, Scottish Secretary in Parliament, has already said the Government will reject the request.
Ms Sturgeon said there appeared to be "no rational reason" to deny a vote.
Downing Street confirmed that the letter was received by email, and said a response would be sent in due course.
"The Scottish government's mandate for offering this choice is beyond doubt", she said.
Sturgeon, who leads the Scottish National Party, complained that her efforts to seek a compromise on Britain's exit from the European Union were met with intransigence from May's Conservative government. But Sturgeon says Brexit has altered conditions dramatically.