Mr Justice Gage asked the Old Bailey jury to concentrate on the evidence they had heard in court with a "cool head and dispassionate view - use your own good judgment".
He said the BBC TV presenter was extremely popular and widely respected and "no doubt her fiance, family and friends loved her dearly".
Mr Barry George, 41, of Crookham Road, Fulham, west London, denies murdering the TV presenter on 26 April 1999.
Miss Dando was shot through the head with a single bullet outside her home in Gowan Avenue, Fulham.
The judge said there had also been a great deal of media coverage of the case, which may have included details which were not in the evidence before the court.
He told the jury: "Have regard to the evidence. Form your own judgment about witnesses and which evidence is reliable and which is not.
"Decide which witness you believe and which witness you do not. Which witness is accurate and truthful and which is not.
'Do not speculate'
"Do not speculate about what evidence there might have been or be driven into speculation in any way."
Mr Justice Gage said there was bound to be a desire to convict the perpetrator of such a dreadful crime.
But he said: "The temptation is to say someone has to pay. Someone has to be convicted for this grave crime.
"You yourselves must have entertained feelings of shock and horror as the account of the killing has been related to you.
"You will no doubt have sympathy for Mr Farthing (her fiance), her family and friends. That is only natural.
"But shut out of your minds all feelings of emotion, sympathy and/or prejudice for any person in this case.
"Ignore all publicity surrounding this case. Put out of your minds anything you have read or heard about this case outside this court."
The jurors were told their verdict would be eagerly awaited by both the press and the public.
But he reminded them Court One of the Old Bailey "had seen many great trials in its day, memories of which have faded".
He said they should feel under no pressure and added: "You should not allow your decision to be affected in any way by consideration of what you perceive to be the consequence of it.
Mr Justice Gage said it was up to the prosecution to prove Mr George was guilty."
As for the defendant, "he does not have to prove anything".
The case was adjourned until Tuesday, when the judge is expected to conclude his summing up.
The jury is expected to be sent out on either Tuesday or Wednesday.