Christian Gerhartsreiter, 52, will hear his fate in a Los Angeles federal court after he lost a bid last month to delay the process.
He could end up getting life behind bars.
Gerhartsreiter was convicted in April of killing John Sohus, his California landlord, who went missing in February 1985 together with his wife Linda. His remains were found nine years later in the backyard of his home in an upscale Los Angeles neighbourhood.
Prosecutors believe the German also killed Linda Sohus, but he was only charged with one murder. When the couple vanished, Gerhartsreiter was living in a guest house owned by the victim's mother.
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Gerhartsreiter -- who also pretended to be a Hollywood producer and an English aristocrat during his years evading arrest after the killing -- came to the United States more than three decades ago.
For more than 10 years, he lived without a driver's license or bank account, never signed a lease and would not even take a flight for fear he would be recognised.
He was finally arrested in 2008 and jailed while awaiting trial.
In closing arguments in April, prosecutor Habib Balian rejected the defence's claim that Sohus's missing wife Linda could just as easily have killed her husband.
Defence attorney Jeffrey Denner also questioned a key piece of prosecution evidence -- two plastic bags wrapped around Sohus's skull, each from a US university where the German had studied.
Gerhartsreiter would have to be "one of the stupidest murderers in the history of Southern California" if he killed Sohus and then wrapped the dead man's head in bags clearly linked to him.
"How he went through life didn't make him very likeable... But that doesn't mean that he is a killer," he said.
A jury found Gerhartsreiter guilty of first degree murder with malicious intent, and of using a blunt object as a deadly and dangerous weapon to kill.