Ms Atkins had campaigned for Britain to stay in the EU but 70.7 per cent of her constituents backed the successful Leave campaign.
That prompted calls from leading out campaigners that Ms Atkins should contemplate the same decision as PM David Cameron amid claims that she was out of touch with her constituents.
But, in an exclusive statement, Ms Atkins says she ‘respects the result’ of the out vote and stressed she intends to play a part in what she termed the ‘disengagement process’ from the EU.
She also paid tribute to the Prime Minister who has, of course, announced he will resign.
Ms Atkins did not respond directly to the claims made by local Brexit officials but in her statement, she said: “The people of Britain - and Louth and Horncastle - have spoken clearly and I respect that result.
“Although I was in favour of our continued membership of the European Union, mine was just one vote like that of everyone else.
“The task ahead of me now is to play my part in maintaining the United Kingdom, in continuing to rebuild our economy and to represent the expressed views of my constituents in effecting the disengagement of the UK from the EU in the most beneficial manner for all of us.
“It will largely happen under a new Prime Minister and I wish to record my thanks to David Cameron for his leadership and achievements, not least in implementing the pledge to let the voters decide on our future in Europe.
“They have now done so and I shall do my very best to abide by their wishes.”