Beggs was convicted in 2001 of murdering 18-year-old Barry Wallace in Kilmarnock, Ayrshire.
The victim's limbs and torso were recovered from Loch Lomond, while his head was found on a beach near Troon.
Beggs was jailed for life and ordered to spend at least 20 years behind bars. A full appeal against his conviction got under way last September, in which he claimed that the trial which saw him convicted of the December 1999 murder was unfair and that he was a victim of a miscarriage of justice. He claimed he was denied a fair hearing because of "prejudicial" publicity before and during the high-profile case.
However, appeal court judges in Edinburgh rejected his appeal in March this year following a legal fight spanning nine years. He then sought leave to appeal to the Supreme Court, which was rejected.
However, Beggs, 47, made a direct application himself to the Supreme Court for special leave to appeal, which has now been rejected.