As International Olympic Committee (IOC) inspectors began their last visit before the Games, the Prime Minister also warned that any Syrian Olympic officials who have been targeted by sanctions against the regime would not be welcome.
More than 7,500 people have been killed in the year-long conflict between Syrian forces and rebels, according to United Nations estimates, sparking concern about whether officials close to the regime should be allowed at the London 2012 Games.
Mr Cameron said Britain is leading the global efforts to seize asset bases and push for sanctions against the regime and many close to President Bashar al-Assad have been banned from travelling within the EU, including his British-born wife Asma al-Assad.
However the Prime Minister told a 10 Downing Street press conference: "I don't think we should punish the athletes for the sins of the region."
He added: "Syrian athletes will be taking part in the Games, and that is right, but let's be absolutely clear: Britain has led efforts within the European Union and elsewhere to institute asset bans, travel freezes and punishing sanctions against this despicable regime and anyone covered by one of those travel bans will not be welcomed in London."
Syria's national committee, headed by General Mowaffak Joumaa who is believed to be a friend of President Assad, would automatically be invited to attend the opening ceremony.
Around eight Syrian athletes could be heading for London but IOC president Jacques Rogge has previously warned that any implicated in war crimes will be banned from competition.
At a press conference with Mr Cameron, Mr Rogge dismissed criticism that London 2012 is being suspiciously secretive over ticket sales.
Calls have been made for a breakdown over tickets in terms of price and by events so a statistical analysis could be made on whether London 2012 was keeping its promises to make seats affordable.
Describing London 2012 as a "transparent company", Mr Rogge said: "This is something that will obviously be fully disclosed soon as the overlay has been finalised and the temporary venues have been constructed.
The draw for the football matches will also have a "major effect" on ticket sales as supporters will want to know where their team will play.
London 2012 has sold seven million Olympic and Paralympic tickets to the public and has another four million to go.
Another million Olympic tickets will go on sale in April. More than a million football tickets and remaining Paralympics tickets still have to be sold.
Mr Rogge said he was confident that London 2012 "will be a great Games", adding: "We are a happy International Olympic Committee."
Mr Cameron responded that "if you are happy, then I am happy" while London 2012 chairman Lord Coe said he was "delirious", only to be jokingly told by Mr Cameron that this is not necessarily the same thing.
The Government and the London 2012 organisers (Locog) have been keen to stress to both the public and the IOC that the £9.3 billion Olympic project is in good hands.
Mr Rogge got an update on Games preparations from Mr Cameron, Culture Secretary Jeremy Hunt, London 2012 chairman Lord Coe and Sports Minister Hugh Robertson. Legacy, transport and security were the topics discussed.
Of the transport and security, which could easily derail plans and positive memories of the Games if an incident were to occur, Mr Rogge said: "We have no doubts whatsoever in these two fields."
Despite mounting a security operation that would be the biggest in British peacetime, Mr Cameron hoped the security measures would not detract from people's enjoyment of the sports events.
Upgrades have been made on the public transport network to try and help avoid travel gridlock and chaos, Mr Cameron said.
Mr Cameron said that the London Games would provide a lasting legacy to improve Britain's sport, health, culture and economy.
London 2012's legacy plans were hailed by Mr Rogge as a "blueprint" for future Games hosts.
It included a focus on sport across all levels and maintaining investment in the Games amid the economic crisis.
Trying to trigger public excitement and community involvement, through the torch relay and recruitment of 70,000 volunteers, plus regenerating London's deprived east end have also been key.
There should be no surprises at this stage of planning for the Games. Mr Rogge described it as "the final straight in a seven-year race" since London were named as the host in 2005.
Backroom talks and presentations with the IOC co-ordination commission of inspectors during the three-day check-up is focussing on the fine detail.
The opening day of the Games, complete with the heads of state, officials, athletes and spectators, both in and outside the Olympic Stadium, is set to be huge test of London 2012's planning skills.
International Paralympic Committee president Sir Philip Craven, who was also at Downing Street, said: "It was a pleasure to meet the Prime Minister to discuss the final preparations for the London 2012 Paralympic Games. There was a real sense of excitement and anticipation in the air.
"The British Government has always been extremely supportive of the Paralympic Games and it was pleasing to hear Mr Cameron talk about how the Games are helping to improve accessibility in the capital and the importance of Paralympians in helping to deliver London 2012's legacy, especially in terms of inclusion."
Mr Cameron said: "The Olympics and Paralympics will be the greatest show on earth. It's an extraordinary honour for the UK and we intend to repay that faith.
"I believe the message is clear: London will be ready, on time and on budget."
He added: "The true legacy of London 2012 lies in the future. Though much has been done, I am acutely aware that the drive to embed and secure the benefits of London 2012 is still to come.
"That is our biggest challenge. It's also our greatest opportunity."
British Olympic Association chairman Lord Moynihan said: "As indicated today by the Prime Minister, the next challenge is to make certain the Games serve as a catalyst for a significant expansion in opportunities to participate in sport throughout the United Kingdom.
"It is the view of the British Olympic Association that every student in the UK should have access to intra and inter-school sport programmes, supported by quality coaching, suitable competition and training venues, a network of well-managed sport clubs, and enthusiastic and knowledgeable volunteers."
PA
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