Marc Webley, 38, was shot outside the Anchor Inn pub in Granton where he was celebrating with friends, it is understood.
He was convicted of being involved in a gangland shooting in 2005.
Police were called to the scene in Granton Crescent at about 11.50pm and he was taken to hospital, along with a 39-year-old man who was also injured.
In a statement on Tuesday, Police Scotland said a red Hyundai Tucson pulled up beside the men in West Granton Road.
The driver got out and fired the gun before getting back in and heading towards Leith.
Detectives believe there was at least one other person in the car at the time.
Police said the shooting is an “isolated targeted attack” and that “this type of reckless criminality will not be tolerated”.
Detective Superintendent Graham Grant said: “An extensive police investigation is ongoing to establish the full circumstances surrounding this incident and we are doing everything we possibly can to identify and trace whoever is responsible.
“We would like to thank members of the public who have assisted our inquiries so far and would continue to ask anyone with information who hasn’t yet spoken to us to please come forward.
“In particular, we are appealing for any information or footage of the suspect vehicle and its movements before and after the incident happened. If you saw this vehicle in the surrounding area late on Sunday evening, we would like to hear from you.
“Our thoughts and condolences remain with the family and friends of both men involved.”
Police said the 39-year-old man remains in a serious but stable condition in hospital.
Superintendent Sam Ainslie said: “We completely understand local community concern following this shocking incident and would like to reassure members of the public that this is believed to have been an isolated targeted attack.
“Police Scotland takes incidents of this nature extremely seriously, this type of reckless criminality will not be tolerated, and we will do everything in our power to bring the perpetrators to justice.
“Officers continue to carry out extra patrols in the area to provide reassurance and we would urge anyone with concerns or information to contact us via 101 quoting reference 3819 of 31 December. Alternatively, Crimestoppers can be contacted anonymously on 0800 555 111.”
The force has set up an online portal to allow people to send information directly to the police at https://mipp.police.uk/operation/SCOT23S46-PO1
Police have not officially released the dead man’s name.
Read More
Those paying tribute to him included his former partner, Jane Park, the Edinburgh EuroMillions lottery winner, who said she is “devastated”.
Ms Park won £1 million on the EuroMillions when she was 17 in 2013.
Parts of Granton Road and Granton Crescent remain closed for police investigations.
In 2005, Webley and and James Tant were each sentenced to 11 years at Glasgow High Court for attempted murder in Edinburgh earlier that year.
Webley, who was 19 at the time, shot another man in Granton in what was described as a failed gangland hit.
