UPDATE: The EDL's Tommy Robinson has said he will not join the BNP on any march.
Nick Griffin's party had initially applied to hold a march and rally from Woolwich Barracks to the Lewisham Islamic Centre.
But concerns over the number of possible counter protests have prompted Scotland Yard to act.
British National Party leader Nick Griffin arrives to lay flowers close to the scene where Drummer Lee Rigby was killed
Griffin has called on his party's supporters to defy the ban and march through Woolwich, even reaching out to the English Defence League's Tommy Robinson to join.
Unite Against Facism have previously said they would form a human shield to stop BNP marchers getting close to Lewisham Islamic Centre.
Conditions have been imposed under the Public Order Act, meaning the BNP march and rally must take place between 1pm and 4pm on Saturday, and can only be held between Old Palace Yard and the Cenotaph in Whitehall.
Police said the information available to them meant "it was necessary to impose the conditions to prevent the demonstration from resulting in serious disorder, serious damage to property, and/or serious disruption to the life of the community".
Police said they had tried to negotiate with the BNP about the precise details of the rally, but were forced to turn to legislation when talks broke down.
Commander Simon Letchford said he was not involved in the discussions so could not comment on them, but said there were "some real concerns" in the community about the rally passing through Woolwich.
Police also said the tight controls were imposed based on "current community tensions, the current intelligence picture about Saturday and recent marches and protests held by similar groups".
Letchford said: "Intelligence was telling us that if this (rally) took place against the backdrop of this tragedy - there were some real concerns.
"I would not say that anywhere is 'off-limits', it's a set of circumstances that if they (the BNP) had conducted their rally in Greenwich at this time, we had a real fear that it could lead to violence and disorder. There's a real risk of it turning ugly.
"Our responsibility is to keep crime off the streets."
The Met said they believed in free speech, and that the action was "proportionate".
Letchford said police would deal with each incident "on its own merits" if any splinter groups decided to march in Woolwich.
He said: "I think our message is if you want to protest, we would ask that you do so peacefully and work with the police to enable that to go ahead."
The ban came as Nick Clegg warned against "knee-jerk" reactions to the murder of Drummer Rigby as he restated his opposition to the so-called "snooper's charter" and stated he does not believe the BNP and Islamists like Anjem Choudary should be banned from TV screens.
The Deputy Prime Minister said measures in the Communications Data Bill were "disproportionate" and "unworkable", despite claims from Cabinet colleagues that the legislation was necessary to ensure public safety.
He also warned that any measures to ban radicals such as the Islam4UK leader from TV screens would make them heroes to extremist groups.
The brutal murder prompted calls from Home Secretary Theresa May and Defence Secretary Philip Hammond for Communications Data Bill legislation to be resurrected.
The controversial legislation would require internet companies to retain records of emails and social media messages for a year and allow police and security agencies to access the data, but not the content of messages.
But speaking on his regular phone-in show on LBC 97.3 today, Clegg said: "We have got to react in a calm way but also a forensic way in deciding exactly what we can do to stop that kind of radicalisation, extremism taking root in individuals and communities."
He insisted he was not seeking to limit powers available to the police and security services.
"I have never suggested that the very considerable powers that our security services and the police have - far in excess, by the way, of many other forces in other parts of the world - should in any way be rolled back."
Clegg said mainstream Muslim groups were "furious" that Anjem Choudary, former leader of banned Islamic group Al Muhajiroun, was given airtime in the wake of the murder.
The Deputy Prime Minister said the signal had to go out to young Muslim men "who might be susceptible to some of these vile, perverted ideologies" that such views were "not an acceptable or recognised understanding of their faith of Islam, it's a total perversion and corruption of it".
But he said it was "not for politicians to tell broadcasters who they can invite on their programmes".
He added: "If you were to say the Government are going to go after this guy, we are going to stop him appearing on British television shows, guess what would happen in my view? He would become a hero in his own community.
"Far from actually discrediting his appalling ideology you would actually turn him into a figurehead. Is that smart? I don't think it's smart."
Earlier on HuffPost:
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