One of the pair also claimed almost £200,000 in benefits by saying the victim – Margaret Fleming – was still alive.
Ms Fleming, who had learning difficulties and was described at the High Court in Glasgow as “friendless and lonely”, would now be aged in her late 30s.
The last independent sighting of her is said to have been at a family event on 17 December 1999.
Her supposed carers Edward Cairney, 77, and Avril Jones, 59, were found guilty of murdering her by unknown means between 18 December 1999 and 5 January 2000 at their home in Inverkip, Inverclyde, or elsewhere in Scotland.
They “literally got away with murder for 16 years”, the Crown said.
Ms Fleming was “manipulated, abused, neglected and ultimately murdered by the two people who should have been looking after her”, Detective Superintendent Paul Livingstone said.
Jones was also convicted unanimously of fraudulently claiming £182,000 in benefits.
A police investigation did not begin until it became clear – in October 2016 – that Ms Fleming was missing, after routine social services inquiries sparked concern.
Ms Fleming went to live with Cairney and Jones when she was a teenager following the death of her father, who “didn’t want her”, the court heard.
The household was said to be in financial difficulties, and by October 1999, various benefits for Ms Fleming were coming in.
The court heard that in correspondence with benefits officials, Jones claimed Ms Fleming had “fantastical” illnesses and conditions.
As police investigated the couple, they claimed she was working as a gangmaster and capable of travelling overseas – as well as being someone with major difficulties in need of a number of benefits.
The Crown claimed the money was “tempting” for the couple but not the “inconvenience” of looking after her Ms Fleming.
Money was the motive for the crime, the court heard, with Cairney and Jones cooking up an “elaborate scheme” to conceal Ms Fleming’s disappearance.
Specialist search teams scoured the cottage where Ms Fleming lived and excavated its grounds.
The defence said it remains a case without a body and without a crime scene.
Detective Superintendent Livingstone said Ms Fleming was “described as being a funny, caring young woman who, despite having some mild learning difficulties, just wanted to be liked and to have friends”.
He added: “The treatment which she was subjected to can only be described as horrific and the conditions in which she lived were utterly disgusting and uninhabitable.
“For Cairney and Jones to continue the charade that she was still alive for all these years is abhorrent, with one of their reasons for doing so being for financial gain.
“We will never know just how Margaret was killed. What we do know is that she lived her last days in what can only be described as a living hell. She must have felt that she was alone in the world with no one coming to help her which
is just heartbreaking to think of.”
Cairney and Jones will be sentenced next month.