Special prosecutor Stan Levco made the request Thursday after both sides presented opening arguments.
Levco claimed defense attorneys violated an agreement not to discuss Camm's two previous trials in the slayings.
Camm has been convicted twice in the shooting deaths at the family's Georgetown, Indiana home.
Both convictions were overturned on appeal.
Camm's current trial is being held in Boone County in central Indiana.
A second man accused in the crime, Charles Boney, was convicted several years ago and is serving 225 years.