Court administrators are taking applications for the job left open by 4th District Court Judge Ray Harding, who pleaded guilty to drug possession charges.
The Administrative Office of the Courts announced Friday it was accepting applications until April 10 for the $103,000-a-year judgeship.
The Judicial Council will interview and narrow down candidates, sending three to five names to Gov. Mike Leavitt.
Leavitt will have 30 days to choose and ask the Senate for confirmation.
The Senate will have two months to act.
Harding, 49, took a plea bargain on Tuesday to a pair of misdemeanor charges of drug possession. He had been charged with felony possession of cocaine and heroin after police responded July 13 to a domestic call made by his wife.
Third District Court Judge Timothy Hanson set sentencing for April 28 on charges that could bring Harding two years in jail and fines of $5,000. He is expected to get probation.
Harding completed a three-month drug treatment program at the Betty Ford Clinic in California last October.
His resignation pre-empted a vote of impeachment at the Utah House of Representatives.
"I hope and pray others who have suffered from addictions and the disease of alcoholism may be able to find hope from the mistakes I've made and the recovery I'm pursuing," Harding said Tuesday.