Cooley replaced former bowling coach David Saker, who resigned in February, for the recent tour of India and the ongoing tour of the United Arab Emirates, where Australia are midway through a one-day international series against Pakistan.
Governing body Cricket Australia confirmed 53-year-old Cooley's Ashes appointment on Monday.
Cooley came to prominence after being appointed England's bowling coach for the 2005 Ashes and was credited for helping Michael Vaughan's bowlers win the urn for the first time since 1986/87.
Australia currently hold the Ashes after beating Joe Root's England 4-0 in the 2017/18 series on home soil.
The 2019 series will be held in England in August and September.
Since replacing Saker, Cooley has overseen a turnaround in Australia's one-day international fortunes, with Aaron Finch's team riding a five-match winning streak which included the recent 3-2 series win over Virat Kohli's India.
Australia lead 2-0 in the five-game ODI series against Pakistan in the UAE.
Cricket Australia added that Tasmania head coach Adam Griffith has been appointed bowling coach for the World Cup in England, reuniting him with head coach Justin Langer.
Griffith has experience working with Australia as an assistant coach on the 2016 tour of the West Indies and was also an assistant coach to former boss Langer at Western Australia.
"I've worked with Adam in the past at the WACA and been pleased with how hard he's worked to establish himself as one of the best young coaches in Australia," Langer said in a media release.
"(Troy) has been a fantastic addition to our coaching team, and brings a great wealth of knowledge, experience and success having coached Australian and England teams in England."
"At the completion of the World Cup and Ashes Series we'll review the overall coaching structure and assess the best path moving forward based on the upcoming schedule, player needs and team direction under Justin," Cricket Australia's interim performance chief Belinda Clark said.
The Cricket World Cup, being hosted by England and Wales, begins on May 30.
(Reporting by Ian Ransom; Editing by Peter Rutherford)
