Relatives, who met David Cameron and Secretary of State Owen Paterson in London on Tuesday afternoon, were expecting to hear that a public inquiry would be held into his death.
However, an 18-month long QC-led review into the circumstances of his murder has been offered instead, with no participation from the family.
Mr Finucane was gunned down by loyalist paramilitaries in front of his wife and children in his north Belfast home in 1989.
The family wanted a Bloody Sunday-style independent inquiry into allegations of collusion between the killers and security forces.
Mr Finucane's widow, Geraldine, says she was insulted by what was offered by the PM and called a halt to the meeting.
'I can barely speak to the media on this occasion, I am so angry," she told reporters outside 10 Downing Street.
I am so angry and so insulted by being brought to Downing Street today to hear what the Prime Minister had on offer.
Mrs Finucane, who said she was hopeful and optimistic before the meeting, added she could not understand how "yet another review of papers" could be justified.
"He is offering a review. He wants a QC to read the papers in my husband's case and that is how he expects to reach the truth," she said.
Retired Canadian judge Peter Cory, who examined allegations of collusion surrounding the Finucane case and other controversial killings at the request of the British and Irish Governments, recommended a public inquiry into the death.
Mr Finucane's son, Michael, who also attended Tuesday's meeting, accused the Prime Minister of "reneging on a commitment that the previous government made to hold a public inquiry".
He said that Mr Cameron had given the "feeble" explanation that public inquiries had not worked in similar cases.
"He seemed oblivious to the fact that the absence of participation by our family would mean we simply couldn't support what he proposed," he added.
The family said they would continue their campaign for a full independent public inquiry.
A Downing Street spokesman said that Mr Cameron told the family that investigations by Judge Cory and John Stevens, then deputy chief constable of Cambridgeshire Police, demonstrated there had been state collusion in the murder.
The Prime Minister expressed his profound sympathy for the family and said it was clear from Stevens and Cory that state collusion had taken place in Mr Finucane's murder.
Downing Street spokesman
"He accepted these conclusions and on behalf of the Government he apologised to the family," the spokesman added.
"He confirmed that the Government's priority was to get to the truth in the best and most effective way.
"The Secretary of State for Northern Ireland will set out the details of this process shortly," he said.
Meanwhile, the Irish Government has pledged to support the family in their bid to obtain an inquiry.
Speaking in the Dáil, Taoiseach Enda Kenny said that if Mrs Finucane was not happy with the outcome of her meeting with the Prime Minister, then the house would not be happy either.
He was responding to comments from Sinn Féin president Gerry Adams.
Mr Adams said Mr Cameron's decision flew "in the face of the huge support given to the family by the (Irish) government, by the UN, by Amnesty International and a host of international agencies and political leaders, including in Washington."
Mr Kenny said Tanaiste Eamon Gilmore will issue an invitation to Mrs Finucane to meet the Government next week to discuss how to move forward.
"I respect the Prime Minister's wish that the truth should be arrived at in the shortest time and that an apology should go to those who are bereaved of the loss of their father and husband," said Mr Kenny.
There is no price you can put on the truth.
Taoiseach Enda Kenny
Meanwhile, SDLP justice spokesperson Alban Maginness described the decision as unacceptable.
The North Belfast MLA said: "After all this length of time one would have expected better from the British Government and Prime Minister on an issue that runs deep into the British military and security complex.
Mr Maginness said he wondered why the Prime Minister had "gone out of his way to raise expectations of a proper inquiry only to dash those hopes with his offer of a review."
However DUP North Belfast MP Nigel Dodds welcomed the outcome.
"Everyone understands the desire of relatives to get the full facts about the death of their loved one," Mr Dodds said.
"However, history in Northern Ireland has shown that the kind of expensive open-ended inquiry demanded in some cases has not been able to bring closure for anyone involved and has actually increased community tensions."
He said that it is "extremely unlikely" that a satisfactory outcome could ever be achieved and the offer put forward by the Government is "entirely reasonable."
UUP victims spokesman Mike Nesbitt said: "I accept the family are left deeply frustrated by today's decision, but it was right to reach a decision and stop the costly impasse.
"What we need is an agreed mechanism, rather than a series of processes that serve to re-write history, painting the state and the agents of the state as the villains."
Alliance Justice Spokesperson Stewart Dickson said: "We need talks to be urgently convened between the parties here and the British and Irish governments to find a coherent way forward to deal with the legacy of the Troubles. "