“We will see the world’s best players competing for the Webb Ellis cup here, where the game was born 200 years ago.
“As a passionate rugby fan, I will share the rollercoaster of emotions with millions of other fans around the world between now and the 31st of October.
“It is this rollercoaster that makes sport, especially rugby, so compelling.”
He added: “I have no doubt that all the highs and lows, dramas and tensions will create unforgettable memories.”
The celebrations were held at the Foreign and Commonwealth office, which the Prime Minister described as a “splendid building”.
He joked: “You can fit number 10 Downing Street in here about five times over”, adding, “I’m going to arrange a job swap one of these days”.
The Prime Minister also revealed that Martin Johnson had made a lasting impression when he earlier visited Downing Street, damaging a 200-year-old chair.
Mr Cameron said he had offered Johnson an “important seat” but when he sat back “the thing snapped in two”.
When Johnson asked if it was an “old chair”, the Prime Minister replied: “Well it had been sat here for 200 years but I’m sure we can find another one”.
The World Cup tournament will feature teams from 20 nations and is the first England has hosted since 1991.
Mr Cameron said he hoped the competition would have a lasting impact on the British public.
“What we are doing here these next six weeks will change people’s lives”, he said, “It can inspire a new generation and sport can be a tremendous power for social movement”.
Harry, a patron of the England Rugby All Schools Programme and vice-patron of the Rugby Football Union (RFU), said the RFU was “committed to harnessing the impact of this competition and the power of rugby for social change”.
He said: “I believe that this World Cup – more than any before – has the potential to grow the game, both at home and abroad, by inspiring children of all backgrounds, and encouraging men and women of all ages and abilities, to embrace rugby.”
He added: “The values of the game, demonstrated so brilliantly by elite players on the global stage, can be mirrored at all levels – both on and off the pitch – to improve lives and strengthen communities.
“I know everyone here shares our enthusiasm for this vision and will join us in helping to create a truly positive and inclusive legacy of this World Cup.”
Harry chatted and joked with eminent rugby figures following his address, including New Zealand coach and former player Pat Lam, who said talking to the Prince was “like chatting to your best mate”.
Lam added Harry “had agreed” that a clash between England and New Zealand would make “a great final”.
RFU president Jason Leonard – a member of the England 2003 World Cup winning squad – said he “firmly” believed the tournament would be “the best ever rugby world cup”, adding that he was committed to ensuring it has a “lasting legacy”.