As is its custom, the high court gave no reasons for denying leave to Fred Prosser.
Prosser was convicted in December 2012 of first-degree murder, sexual assault causing bodily harm, and sexual assault in the death of Sabrina Patterson.
Patterson's body was found near Shenstone, N.B., in October 2010 about a week after she went missing.
The 25-year-old woman had been strangled and was partially wrapped in a garbage bag that had been left by a road in a wooded area.
Prosser was making a last attempt to appeal his conviction, after his lawyers previously argued there were errors in his case, including with DNA evidence.