Detectives fear the "best opportunity" to finally bring Pc Yvonne Fletcher's killers to justice has gone after evidence which identified them was withheld for national security reasons.
The 25-year-old's family said they are "deeply disappointed and frustrated" a Libyan national suspected of conspiring in her murder will not be prosecuted.
The suspect, who is in his 50s, has been released from bail without charge and told the investigation against him will not proceed "at this time", the Metropolitan Police said.
A decision that crucial material cannot be used in court all but marks the end of the 33-year hunt for her murderers as the force admitted the likelihood of another breakthrough is "low".
The Police Federation of England and Wales said the decision was a "disgrace", while Home Secretary Amber Rudd praised detectives but declined to comment on the details of the case.
Pc Fletcher was shot by a sniper as she policed a demonstration against Colonel Muammar Gaddafi outside the Libyan People's Bureau in St James's Square on April 17 1984.
The shooting sparked a 10-day siege of the building before 30 of the occupants were deported back to Libya.
No-one has ever been prosecuted over the killing and Pc Fletcher's father Tim died with "one regret" that he never saw justice.
Her family said they had hoped progress made by counter-terrorism officers would provide them with answers.
In a statement, they said: "We understand that some available evidence could not be used in court but are satisfied that the Metropolitan Police has left no stone unturned in its pursuit of justice in Yvonne's case.
"We are deeply disappointed and frustrated that a prosecution cannot proceed at this time.
"We had hoped that the latest turn of events would finally lead to some closure for the family."
The suspect was arrested for conspiracy to murder, alongside two others held for money laundering, in south-east England in November 2015 in what police then described as a "significant turning point" in the inquiry.
The demise of Gaddafi - who was killed in 2011 during civil war in Libya - and the regime change that followed had granted them access to pursue new leads.
But the force said that despite detectives "tirelessly" pursuing hundreds of lines of inquiry, including visits to Libya, the previously unseen evidence they gathered is adjudged to be inadmissible in court.
A spokesman for the Met said: "We believe our investigation has identified enough material to identify those responsible for WPc Fletcher's murder if it could be presented to a court.
"However, the key material has not been made available for use in court in evidential form for reasons of national security.
"Therefore, without this material and following a review of all the evidence that was available to prosecutors, the Crown Prosecution Service, who we worked closely with throughout, have informed us that there is insufficient admissible evidence to charge the man."
It has not been disclosed what the nature of the material is or who barred its use in open court.
The Met statement added the investigation into the "act of state-sponsored terrorism" remains open but it is unlikely more evidence will be found.
The force said: "Our judgment is that this concludes what was by far the best opportunity to solve this tragic case and provide a degree of closure for the victims and their families.
"This investigation will never be closed but the likelihood of finding further evidence, in Libya or elsewhere, is low."
Assistant Commissioner Mark Rowley said: "I regret that we have not been able to deliver the justice that the victims and their families deserve."
Ms Rudd praised the "hard work" of investigators.
She said: "WPc Fletcher was one of their own.
"Her murder remains as shocking and senseless as the day it occurred and I understand that the decision will be deeply disappointing and frustrating for all her family, friends and colleagues."
A Foreign Office spokesman said: "We do not routinely comment on intelligence matters."
Steve White, chairman of the Police Federation of England and Wales, said the decision was a disgrace.
"Can the law really be fit for purpose when it protects those responsible for killing someone charged with keeping us safe?" he said.
"The Metropolitan Police have been scrupulous in their efforts to bring Yvonne's killer to justice but this will look to many like people can quite literally get away with murder, it's a disgrace.
"We will never forget Yvonne or any of those that pay the ultimate sacrifice."