Published Monday, 30 September 2013
The 17-year-old from Donaghmore, Co Tyrone, was abducted and murdered by the IRA while working in Dublin in November 1975.
Teams from the Commission for the Location of Victims' Remains (ICLVR) had been looking for his body at Bragan Bog, near Tydavnet, Co Monaghan.
A statement from the organisation on Monday said the search has been completed.
It said: "Regrettably Mr McVeigh's remains have not yet been found. This was the sixth of a series of searches for Columba McVeigh undertaken by the Commission in this location since 1999.
"A total of 14.25 acres (5.5 hectares) of ground has been excavated and searched."
There were 16 people who disappeared during the Troubles in Northern Ireland. The remains of nine of them have been found while seven are still to be located.
Why do they continue to torture a family like this?
The search for Columba McVeigh had been taking place since April close to an area of bogland where previous searches were halted in Autumn 2012, due to bad weather.
In an emotional plea for fresh information, his family urged anyone who can to come forward with details that could end their "torture".
Columba's brother, Oliver McVeigh, said: "The news that the sixth search of Bragan has not found Columba is devastating for the family. The ICLVR is doing everything it can to find Columba but it's not going to happen unless they get accurate information.
"There are people out there who have the information and they haven't come forward and I can't understand why. They have nothing to gain by keeping their secret and nothing to lose by telling the Commission what they know. They know that the information is confidential."
The family also released the last letter Columba sent to his mother, in which his final words were: "Well I'll say bye for now. God bless and take care. From your big son, Columba."
His sister Dympna said: "Our mother went to her grave unable to tend the grave of her son.
"If the people who know where Columba is buried could have seen what that did to my mother, if they could imagine their own mother in that position they could not stay silent if they had any human feelings at all.
"I would like to look them in the eye and ask them: 'Do you know what you have done'? Let us bring Columba home and end this torment".
Anyone with information can pass it on to the commission on 00800 5558 5500.
© UTV News