The broadcast was specially recorded at Windsor Castle
She said the outpouring of affection over the last few days had been "overwhelming".
"I have drawn great comfort from so many individual acts of kindness and respect," she said.
Queen Mother's funeral
Find out more She revealed her sadness in an address recorded earlier on Monday, the day before the Queen Mother's funeral at Westminster Abbey.
Speaking at Windsor Castle, the Queen gave thanks for her mother's long and eventful life.
"She had an infectious zest for living, and this remained with her until the very end," the Queen said.
But she said she hoped that at Tuesday's ceremony sadness would blend with a wider sense of thanksgiving for her mother's life and for the times in which she lived.
Her death had, she added, left a "void" in the family's life.
Shortly before the address was carried across Britain's TV and radio stations and online, the Queen Mother's four grandsons, princes Charles, Andrew, and Edward, and Viscount Linley, held a vigil beside her coffin.
The Queen's broadcast at 1800 BST also came after other members of the Royal Family went on an impromptu walkabout among the crowds of thousands outside Westminster Hall.
Silent vigil
Queues of people lined the streets to pay their respects to the Queen Mother as her coffin lay in state.
The grandsons remained on silent guard for 20 minutes.
Prince William and Harry were inside the hall to support their father and pay their own respects.
The Royal party also shook hands with people in the crowd during the walk along the South Bank of the river Thames.
The Prince of Wales later paid a second visit to Westminster Hall to pay further respects to his grandmother.
Charles quietly slipped in to the Great Hall and stood on the VIP dais at around 2300 BST.
Dressed in a black suit and black tie, he spent around 20 minutes in sombre silence as he looked over his grandmother's coffin.
Some members of the public have already camped out ahead of the funeral.
Two minute silence
The service will be relayed on speakers along The Mall and outside Westminster Abbey on North Green.
The Queen Mother will finally be laid to rest on Tuesday evening at Windsor, alongside the body of her late husband King George VI and the ashes of her daughter Princess Margaret.
Black Rod, Sir Michael Willcocks, said more than 160,000 people had paid their respects at Westminster Hall so far, with well-wishers passing through at a rate of 5,000 per hour currently.
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The tenor bell at Westminster Abbey will toll every minute for 101 minutes before the service.
There will be a two-minute silence in many offices and shops at 1130 BST (1030 GMT) at the beginning of the funeral.
Afterwards, a procession to Windsor will begin with a fly-past tribute by two Spitfires and Britain's only remaining airworthy Lancaster bomber, as the cortege moves up The Mall.
Westminster Hall will close at 0600 BST on Tuesday to allow funeral preparations to begin.
Sir Michael Westminster Hall said at some point in the early hours, officials would have to start informing people that it was too late to join the queue.