The Stack family say they have received an apology from Garda Commissioner Drew Harris in relation to the handling of the investigation into the IRA murder of prison officer Brian Stack in 1983.
Speaking after a meeting at Garda headquarters on Wednesday, Austin Stack said gardai will submit a file to the DPP but do not believe there is sufficient evidence to bring a prosecution in the case.
Brian Stack was gunned down by the IRA as he left an amateur boxing match in the National Stadium in Dublin in March, 1983. He died of his injuries 18 months later.
Speaking to the Irish Independent ahead of the meeting, Austin Stack said his family would be seeking a full apology from Commissioner Harris over the mishandling of the original Garda investigation into his father's murder.
"We will be expecting a full, meaningful and sincere apology for the mishandling of the original investigation into our father's murder," he said.
In a statement, the family said they want Justice Minister Charlie Flanagan to appoint an expert in policing from outside the jurisdiction to investigate failings in the garda investigation.
"We have got an apology from Commissioner Harris. We have accepted it. However, we have to say that it has been a long time coming, but at least it is a start."
Mr Stack's wife Shelia and her other two sons Oliver and Kieran were also due to attend.
Gardai released an official statement from Garda Commissioner Drew Harris shortly after the meeting;
"Along with senior detectives, I met with the Stack family this morning to provide them with a briefing on the ongoing investigation into the murder of Mr Brian Stack," Mr Harris said.
"I offered the Stack family an apology for the failings and shortcomings in the investigation. I fully acknowledge that these matters are serious and had a detrimental impact on the investigation.
"This investigation remains open and An Garda Siochana would appeal for anyone with information in relation to the murder of Mr Brian Stack to come forward."