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This event is now over, but you can see how it happened below:
The men finally enter the room, exactly half an hour late.
Mr Brown calls Bush "a true friend of Britain".
The two men could not appear any different. Mr Bush is, as ever, smirking slightly. Mr Brown is dour, serious, business-minded and not a little monotone.
Mr Brown says they've discussed Burma, Afghanistan and Zimbabwe.
Mr Bush winks at someone in the audience. You just can't see the prime minister doing something like that. Even the president struggles to look interested in what Gordon Brown is saying.
The prime minister announces more sanctions on Iran.
He calls for free and fair elections in Zimbabwe. The two also announce childbirth, malaria and AIDS projects for the developing world.
Mr Brown moves onto oil and his constant "dialogue between producers and consumers".
"The United States has played an essential role in achieving peace in Northern Ireland," Mr Brown says, setting up the president's trip there this afternoon. President Bush is to be thanked by all the people of the United Kingdom for what he's done, Brown continues.
Bush is on now. "This has been a good trip. Some are speculating this is my last trip. Let them speculate, who knows?"
Now he thanks Brown for having him over to dinner last night, and for letting him meet some historians Brown invited over. The president claims to love reading history books.
Mr Bush says the prime minister is tough on terror "and I appreciate it".
Typically for the two, Mr Bush's words are abstract and binary - freedom, terror, moral duty etc. Mr Brown reels off policies and proposals, barely touching on any sort of good vs evil narrative.
"He's left more troops in Iraq than originally indicated," Mr Bush says about Mr Brown. He also thanks the prime minister for his Afghanistan troop announcement, one he hasn't really made yet.
The words follow an irritated section about media speculation on Gordon Brown for possibly distancing himself from the Americans.
On Iran, Mr Bush thanks Brown for his "strong words". On Zimbabwe he says: "You're obviously emotional on the subject, and I don't blame you."
Mr Bush talks about which countries have matched US aid in Africa and says Britain is one of them. As he says this he touches Mr Brown on the arms. Mr Brown responds to that with one the least convincing smiles in politics.
On climate change Bush talks of an international goal for 2050. This has to be a real area of tension for the two, with US environmental action plans treated with total contempt by environmental groups and Britain going down the road of some of the most wide-ranging climate change legislation in the world.
Nick Robinson of the BBC asks the first question. He and Bush famously despise each other. His relatively tough question (nothing out the ordinary for British press) is met with: "We miss you, Nick," from the president.
The question is about troop levels in the Iraq, with the suggestion Mr Brown is too nervous to tell Mr Bush he wants out. Mr Brown gives a textbook 'we're moving to overwatch' response.
"We're withdrawing troops," Mr Bush says - positing a number of 30,000. He's using this as a demonstration of how successful deployment has been in Iraq and quickly claims the Iraqis are capable of running themselves now. "We're bringing ours home too," he continues.
Brown just can't stand still. He shifts from left foot to right foot, from looking at Bush to looking away, every moment or so. Nevertheless, it's still more satisfying to watch him with Bush than it is Blair. One feels they retain a healthy distance from each other, and one gets a greater feeling of them being equal partners - however false - than one ever did with Tony Blair.
"Is it possible you got it wrong [on Iraq]," a journalist asks Bush.
"History will judge the tactics," Mr Bush replies.
"Absolutely it's necessary if you believe we're in an ideological war," Bush continues. "The strategic implications of a free Iraq are significant for our future."
You can't see Brown saying the phrase 'ideological war', and many people in the room will be grateful for that.
Mr Bush now launches into a defense of his foreign policy in the most conceptual manner imaginable.
Mr Brown talks about Iraqis having an economic stake in their country. Somewhat more down-to-earth, somewhat more boring, but somewhat more in line with what the assembled British journalists are comfortable with.
That being said, president Bush - a man derided as the 'child president' by some on the American left - still outshines Mr Brown in the charisma stake. He at least talks without notes, and appears more comfortable and confident infront of the cameras.
A journalist questioning Mr Bush's policy on Iran recieves an angry response barely on the right side of shouting. As ever, Mr Bush does not enjoy being subjected to hostile questioning.
He struggles to explain the difference between the US building civilian power stations and Iran doing it but eventually explains it's to do with trust. He reiterates Russian plans for the US to give the Iranians the necessary nuclear fuel and then take it away when it's been used.
All of a sudden, he says: "Thank you very much". It couldn't really make the relationship between Brown and Bush any clearer. As soon as Bush decides it's time to leave Mr Brown decides it's time to leave too.
The two men shake hands and walk off the stage.