William Kean was found guilty at the High Court in Glasgow in August of bludgeoning 80-year-old Mrs Methven to death at her rural cottage near Forteviot, Perthshire.
Judge Lord Glennie sentenced him to life in jail and ordered that he serve at least 22 years before being considered for release.
Kean's family insist he is innocent – and he took the first legal steps to clear his name last year. But it emerged yesterday Kean's lawyers have dropped their appeal bid.
A High Court of Justiciary spokesman said the deadline had passed for paperwork to be lodged to continue the legal process to have the case looked at by the Court of Criminal Appeal. They had earlier been given an eight-week extension to the deadline to submit the paperwork to take it further.
A spokeswoman said: "No appeal has been lodged but they could in the future come back and try to get an extension, which would have to go before a judge."
The spokeswoman added that Kean's lawyers, J Myles and Co of Dundee and Carnoustie, had told High Court staff an appeal was not being lodged.
The firm declined to comment or indicate if an appeal could be resurrected at a later date.