A jury at the Old Bailey court in London had failed to reach a verdict on the charge last week while finding him guilty of conspiracy to hack phones.
Coulson, 46, the former News of the World (NoW) editor faces a maximum of two years in prison and is due to be sentenced later this week for plotting to hack phones at the erstwhile Murdoch-owned tabloid between 2000 and 2006.
"The CPS has taken the position to proceed with the retrial," Prosecutor Andrew Edis QC confirmed.
The new trial will deliberate on allegations that Coulson gave approval to Goodman to pay a Buckingham Palace police officer for two royal telephone directories.
Coulson appeared today at the Old Bailey as his sentencing process for the phone-hack ruling began, alongside former colleagues Neville Thurlbeck, Greg Miskiw and James Weatherup who have all admitted their part in what the court heard was "systemic misconduct".
Former NoW reporter Dan Evans, who has also admitted phone-hacking, will be sentenced separately in late July.
Cameron, who apologised for hiring Coulson soon after his conviction, was criticised by the judge as the jury were still considering the other two charges.
He accused him of launching "open season" on Coulson as the premier accused his former aide of lying to him.
Rebekah Brooks, the former editor of NoW and the 'Sun', was cleared of all charges in the trial, as were her husband Charlie and former PA Cheryl Carter, News International's former head of security Mark Hanna and former NoW managing editor Stuart Kuttner.
Employees of the newspaper were accused of engaging in phone-hacking, police bribery, and exercising improper influence in the pursuit of publishing stories.
The 168-year-old NoW shut down in disgrace in July 2011 amid a public outcry.
The scandal sent tremors across Britain's political elite. Cameron, known to be close to Murdoch and Brooks, was forced to order a public inquiry into press ethics in the immediate aftermath of the raging row.
