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The playwright beat Sir Winston Churchill into second place by a narrow margin with William Caxton a little way behind in third.
The result of the contest was kept a close secret until it was announced at 0815 GMT on the New Year's Day edition of Today.
Over the Christmas period listeners were asked to choose their Person of the Millennium and nominations were reduced to a shortlist of six. The other three on the shortlist were Oliver Cromwell, Charles Darwin and Sir Isaac Newton.
Then more than 45,000 voted by telephone for the eventual winner, the world famous Shakespeare.
Actress Dame Judi Dench has played many Shakespearian roles in her 40-year career.
She said her household was particularly glad Shakespeare had won the accolade "because he is known in our house as 'the gentleman who pays the rent'.
"He paid our rent for many, many years."
Dame Judi said she had played Ophelia in 1957, for which she was "panned by the critics".
She added: "One of the reasons was that I was just out of drama school."
Dame Judi named Twelfth Night as her favourite Shakepeare play: "It's wonderfully constructed, so beautiful."
And referring to her 1987 performance of Cleopatra she said: "I remember a marvellous note that [director] Peter Hall gave me and he said ... 'don't ever think of coming on for one scene and having to be all of her'.
"You come on in one scene and show one aspect of her and another scene maybe another aspect of her and so on. By the end of the evening hopefully you will have the whole person, the whole woman."
Professor Stanley Wells, of the Shakespeare Birthplace Trust, told Today he was "delighted to receive the award on Shakespeare's behalf."
Prof Wells said Shakespeare's plays had not found immediate favour during his own lifetime, and it was not until the 1790s that the Bard's influence had begun to grow.
He added: "In this century indeed it's spread enormously internationally. That's another of the reasons why he's become such a potent world force nowadays that people have been able to take him up - he translates so well."
Prof Wells cited A Midsummer Night's Dream as a favourite but said King Lear was the playwright's greatest play in depth of understanding.
Voting for the final six contenders was:
For UK and international users, there will be further opportunities throughout 1999 to nominate people of the Millennium in a variety of categories on BBC News Online.
News Online launched Your Millennium on New Year's Day. Throughout the year visitors will be asked to vote for their choice of Person of the Millennium in a different category each month - starting this month with the Inventor of the Millennium.
Your Millennium will not be confined to British nominations. Any inventor in the world in the last 1,000 years qualifies.
In February, nominations will be invited for Writer of the Millennium. And at the end of the year, Your Millennium will invite nominations for the Man and Woman of the Millennium.
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