But he will continue to appeal against the minimum of 37 years he was ordered to serve in prison
The sentence was the longest ever passed by a Scottish court.
Sinclair was convicted last year of raping and murdering Christine Eadie and Helen Scott in 1977. The teenagers disappeared after being in the World's End pub in Edinburgh's Royal Mile.
Their naked bodies were later found in a field in East Lothian.
Police believe Sinclair has also killed at least six other women and girls.
The sentence given to Sinclair, who will turn 70 this year, means he would be aged 106 before he was eligible to apply for parole.
His conviction brought one of Scotland's most infamous unsolved cases to a conclusion, and marked the first prosecution since changes to Scotland's double jeopardy law.
Sinclair, a violent offender who has already spent more than half of his life in prison, had intended to appeal against the World's End conviction.
But a procedural hearing at the Appeal Court in Edinburgh was told that he no longer intends to appeal against the conviction, a spokesman for the Judicial Office for Scotland confirmed.
A formal abandonment is expected to be lodged in due course and his appeal against sentence will be heard in November.
Worlds End murderer Angus Sinclair is to drop his appeal against his conviction.