Gary Dobson and David Norris were found guilty by an Old Bailey jury after a trial based on forensic evidence.
Scientists found a tiny bloodstain on Dobson’s jacket that could only have come from Mr Lawrence.
They also found a single hair belonging to the teenager on Norris’s jeans. Sentencing will be on Wednesday.
In an exclusive interview with the BBC’s Panorama, Stephen’s mother Doreen Lawrence said: “I don’t forgive the boys who killed Stephen. They don’t think they have done anything wrong.
“They took away Stephen’s life and there is nothing in their behaviour or anything to show they regret what their actions have done and the pain it has caused us as a family.”
In a statement read by his lawyer outside the Old Bailey, Stephen’s father, Neville Lawrence, said the convictions were a moment of joy and relief – but he could not rest until all of those who killed his son were brought to justice. He described the investigation and preparation of the case as “faultless”.
Duwayne Brooks, Stephen’s best friend who had been with him on the night of the murder, tweeted: “Some justice at last”.
The original failed investigation into the murder led to the Metropolitan Police being branded as institutionally racist.
Stephen Lawrence was 18 when he was stabbed to death near a bus stop in Eltham, south London, in April 1993.
Police identified five men who were later named in a damning public inquiry as the “prime suspects”.
By that time, there had already been a catalogue of police errors and two failed prosecutions, one brought by Stephen’s parents.
But in a four-year-long cold case review, a fresh team of forensic scientists uncovered microscopic evidence linking two of the five men to the murder – evidence that the police had held all along.
The material – bloodstains, clothing fibres and a single hair belonging to the teenager – were recovered from the clothes of the suspects which had been seized in 1993.
Scientists recovered the material using advanced techniques which were not available to the original case scientists.
Dobson, 36, and Norris, 35, denied the murder. They said their clothing had been contaminated as police mixed up evidence over the years. Detectives spent months establishing the movements and handling of the exhibits since 1993 – and the jury were told that contamination was implausible.
Dobson was jailed for five years in 2010 for drugs trafficking. He is among a small number of men to have been tried twice for the same crime after the Court of Appeal quashed his 1996 acquittal for the murder.
Norris was convicted in 2002 of a separate allegation of racially threatening behaviour.
In mitigation, ahead of sentencing on Wednesday, counsel for Dobson said there was no evidence he had been the leader or prime motivator of the group that attacked the teenager.
Norris’s counsel repeated his client’s pleas of innocence – and revealed his client had been beaten up while on remand at Belmarsh prison, suffering a broken nose and four broken ribs.
Mr Justice Treacy discharged the jury and thanked them for their “dedicated service”. He told them the public owed them a debt of gratitude.
As the defendants left the dock, Dobson told his family not to worry, and Norris waved to the gallery. Members of both men’s families shouted back.
Acting Assistant Commissioner Cressida Dick, who ordered the 2006 cold case review that led to the convictions, said the case had been extremely important for the Lawrence family, the Metropolitan Police and society at large.
She said: “It’s a matter of huge regret to the Met that it has taken 18 years to get to this point.
“It has been a unique case in policing. Firstly the horrible, horrible nature of the attack on the night, the time in which it has taken to bring anybody to justice, and the tireless campaigning of the Lawrences.
“There is no comparable case. All homicide cases are terrible, but for us it is a very important case.
“Most importantly, we wanted to be able to bring people to justice for the killing and try to give Doreen and Neville Lawrence and their family some sense of justice.”
source: BBC © 2011