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8 year oldMrs Trump, the third wife of Republican candidate Donald Trump, faced accusations that parts of her speech had plagiarised Michelle Obama.
Lines in the speech matched almost word for word those delivered by Mrs Obama at the Democrats' convention in 2008.
Ms McIver said her offer to resign was rejected by the Trumps.
"This was my mistake, and I feel terrible for the chaos I have caused," she said in a statement.
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The Trump staffer said Mrs Trump had read out passages from Mrs Obama's convention speech in a phone conversation as they discussed people who inspired her.
"A person she has always liked is Michelle Obama," she wrote.
Ms McIver later included the phrases in a draft of the speech without checking Mrs Obama's speeches, she added.
"I am honoured to work for such a great family," she said.
Mr Trump was formally adopted as the Republican candidate for November's presidential election on Tuesday.
In the aftermath of the controversy over his wife's speech, he tweeted that the publicity was good news as the speech "got more publicity than any in the history of politics".
Mr Trump's campaign manager had emphatically denied plagiarism.
Republican strategist Sean Spicer even invoked My Little Pony in an interview with CNN - citing similarities between the words of Twilight Sparkle (a purple unicorn from the children's franchise) and Mrs Trump's speech to show that the lines were "common phrases".
Thirty-six hours later, the Trump campaign issued the press release it could have - should have - sent out in the hours after allegations of plagiarism in Melania Trump's speech first surfaced.
Instead the story festered for more than a day, sucking up precious political oxygen during a week that Republicans hoped would be a showcase for the party.
This will do nothing to assuage the concerns of some conservatives who wonder about the professionalism and competency of the Trump political operation.
The about-face also leaves those who were ardently defending Mrs Trump over the past day and a half - such as campaign chair Paul Manafort - playing the fools.
Meanwhile, Meredith McIver appears to have avoided the axe, despite her offer to resign.
It seems that Mr Trump, famous for dramatically firing people for professional incompetence - he had a reality television show built around the concept, after all - will forgive what is usually viewed as a cardinal sin among speechwriters.
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